From The Code of Canon Law, Book IV:
Can. 844 §1 Catholic ministers may lawfully administer the sacraments only to catholic members of Christ's faithful, who equally may lawfully receive them only from catholic ministers, except as provided in §2, 3 and 4 of this canon and in can. 861 §2.The Catholic Church has great respect for Orthodoxy and we long for the day when the schism can be mended. There are significant current attempts to bring that about. Pope John Paul II wrote an excellent Apostolic Letter in 1995 entitled Orientale Lumen ("The Light of the East").
§2 Whenever necessity requires or a genuine spiritual advantage commends it, and provided the danger of error or indifferentism is avoided, Christ's faithful for whom it is physically or morally impossible to approach a catholic minister, may lawfully receive the sacraments of penance, the Eucharist and anointing of the sick from non-Catholic ministers in whose Churches these sacraments are valid.
§3 Catholic ministers may lawfully administer the sacraments of penance, the Eucharist and anointing of the sick to members of the eastern Churches not in full communion with the catholic Church, if they spontaneously ask for them and are properly disposed. The same applies to members of other Churches which the Apostolic See judges to be in the same position as the aforesaid eastern Churches so far as the sacraments are concerned.
§4 If there is a danger of death or if, in the judgment of the diocesan Bishop or of the Episcopal Conference, there is some other grave and pressing need, catholic ministers may lawfully administer these same sacraments to other Christians not in full communion with the catholic Church, who cannot approach a minister of their own community and who spontaneously ask for them, provided that they demonstrate the catholic faith in respect of these sacraments and are properly disposed.
[Dave: Canon 861 §2 states that in emergency situations, anyone who has the right intention, can baptize]
The Christian tradition of the East implies a way of accepting, understanding and living faith in the Lord Jesus. In this sense it is extremely close to the Christian tradition of the West, which is born of and nourished by the same faith. Yet it is legitimately and admirably distinguished from the latter, since Eastern Christians have their own way of perceiving and understanding, and thus an original way of living their relationship with the Saviour. . . .That is the ecumenical and diplomatic approach. Of course, I myself, being an apologist, have written some material explaining where Catholics and Orthodox have honest differences, and why I am a Catholic and not an Orthodox. See my introductory paper on that topic.
17. Thirty years have passed since the Bishops of the Catholic Church, meeting in Council in the presence of many brothers from other Churches and Ecclesial Communities, listened to the voice of the Spirit as he shed light on deep truths about the nature of the Church, showing that all believers in Christ were far closer than they could imagine, all journeying towards the one Lord, all sustained and supported by his grace. An ever more pressing invitation to unity emerged at that point.
Since then, much ground has been covered in reciprocal knowledge. This has increased our respect and has frequently enabled us to pray to the one Lord together and to pray for one another, on a path of love that is already a pilgrimage of unity.
After the important steps taken by Pope Paul VI, I have wished the path of mutual knowledge in charity to be continued. I can testify to the deep joy that the fraternal meeting with so many heads and representatives of Churches and Ecclesial Communities has given me in recent years. . . .
24. I believe that one important way to grow in mutual understanding and unity consists precisely in improving our knowledge of one another. The children of the Catholic Church already know the ways indicated by the Holy See for achieving this: to know the liturgy of the Eastern Churches;  to deepen their knowledge of the spiritual traditions of the Fathers and doctors of the Christian East,  to follow the example of the Eastern Churches for the inculturation of the Gospel message; to combat tensions between Latins and Orientals and to encourage dialogue between Catholics and the Orthodox; to train in specialized institutions theologians, liturgists, historians and canonists for the Christian East who in turn can spread knowledge of the Eastern; Churches; to offer appropriate teaching on these subjects in seminaries and theological faculties, especially to future priests.  These remain very sound recommendations on which I intend to insist with particular force.
25. In addition to knowledge, I feel that meeting one another regularly is very important. In this regard, I hope that monasteries will make a particular effort, precisely because of the unique role played by monastic life within the Churches and because of the many unifying aspects of the monastic experience, and therefore of spiritual awareness, in the East and in the West. Another form of meeting consists in welcoming Orthodox professors and students to the Pontifical Universities and other Catholic academic institutions. We will continue to do all we can to extend this welcome on a wider scale. May God also bless the founding and development of places designed precisely to offer hospitality to our brothers of the East, including such places in this city of Rome where the living, shared memory of the leaders of the Apostles and of so many martyrs is preserved. . . .
28. . . . [end section]
We are painfully aware that we cannot yet share in the same Eucharist. Now that the millennium is drawing to a close and our gaze turns to the rising Sun, with gratitude we find these men and women before our eyes and in our heart.
The echo of the Gospel--the words that do not disappoint--continues to resound with force, weakened only by our separation: Christ cries out, but man finds it hard to hear his voice, because we fail to speak with one accord. We listen together to the cry of those who want to hear God's entire Word. The words of the West need the words of the East, so that God's word may ever more clearly reveal its unfathomable riches. Our words will meet for ever in the heavenly Jerusalem, but we ask and wish that this meeting be anticipated in the holy Church which is still on her way towards the fullness of the Kingdom.
May God shorten the time and distance. may Christ, the Orientale Lumen, soon, very soon, grant us to discover that in fact, despite so many centuries of distance, we were very close, because together, perhaps without knowing it, we were walking towards the one Lord, and thus towards one another.
May the people of the third millennium be able to enjoy this discovery, finally achieved by a word that is harmonious and thus fully credible, proclaimed by brothers and sisters who love one another and thank one another for the riches which they exchange. Thus shall we offer ourselves to God with the pure hands of reconciliation, and the people of the world will have one more well-founded reason to believe and to hope.
Sometimes, I have been confronted with the sub-group of anti-Catholic Orthodox on the Internet and felt compelled to explain why I thought it was impossible to take a position that Orthodoxy was apostolic while Catholicism supposedly was not (an extreme opinion in these ranks would even hold that Catholicism lacks both sacraments and grace, as well as apostolicity). See my two papers on that (one / two). For more of my papers regarding Orthodoxy, see my Orthodoxy Topical Index page.
As I understand it, the biggest issue at the time of the schism in 1054 was the question of the filioque clause in the western version of the Nicene Creed. I believe the mutual excommunications have been dropped, and with increasing discussion, it can be seen, I believe, that there is far more agreement on this issue than was formerly assumed. A lot of the disagreement had to do with different approaches of eastern and western Christianity, and there were some purely linguistic misunderstandings, too. See the paper from William Klimon on filioque.
As to the Oriental Orthodox, here is what the article in Wikipedia states:
The schism between Oriental Orthodoxy and what would become the Roman Catholic and Eastern Orthodox churches occurred in the 5th century. The separation resulted in part from the refusal of Pope Dioscorus, the patriarch of Alexandria, to accept the Christological dogmas promulgated by the Council of Chalcedon, which held that Jesus has two natures — one divine and one human. This was not because the council stated that Christ has two natures, but because the council's presiders refused to confess (more than wordly) that the two natures are inseparable and united. Pope Dioscorus would accept only "of or from two natures" but not "in two natures."
To the hierarchs who would lead the Oriental Orthodox, this was tantamount to accepting Nestorian-flavored terminology, according their definition of Christology, which was founded in the Alexandrine School of Theology that advocated a formula that stressed unity of the Incarnation over all other considerations.
The Oriental Orthodox churches were therefore often called Monophysite churches, although they reject this label, which is associated with Eutychian Monophysitism, preferring the term "non-Chalcedonian" or "Miaphysite" churches. Oriental Orthodox Churches reject the heretical Monophysite teachings of Eutyches, the heretical teachings of Nestorius and the Dyophysite definition of the Council of Chalcedon.
Christology, although important, was not the only reason for the refusal of the Council of Chalcedon - political, ecclesiastical and imperial issues were hotly debated.
In the years following Chalcedon, the patriarchs of Constantinople remained in communion with the non-Chalcedonian patriarchs of Alexandria, Antioch, Jerusalem, while Rome remained out of communion with Alexandria, Antioch and Jerusalem, and in unstable communion with Constantinople. It was not until 518 AD that the Byzantine Emperor, Justin I, on the ultimatum of the Roman patriarch, demanded that the Church of the Roman Empire be Chalcedonian once and for all. Justin ordered the deposition and replacement of all anti-Chalcedonian bishops, including the patriarchs of Antioch and Alexandria. By 525 AD, anti-Chalcedonian Christians found themselves being persecuted by the Roman Empire; this would not end until the rise of Islam.
In the 20th century, the Chalcedonian schism was not seen with the same relevance any more, and from several meetings between the Roman Catholic Pope and Patriarchs of the Oriental Orthodoxy, reconciling declarations emerged.
Many Eastern Orthodox will freely admit the primacy of Peter himself and even the primacy of the Apostolic Roman See in the early Church. I have a book in my library by prominent Orthodox author John Meyendorff, entitled The Primacy of Peter.
From the common declaration of Pope John Paul II and HH Mar Ignatius Zakka I Iwas, June 23, 1984
“ The confusions and schisms that occurred between their Churches in the later centuries, they realize today, in no way affect or touch the substance of their faith, since these arose only because of differences in terminology and culture and in the various formulae adopted by different theological schools to express the same matter. Accordingly, we find today no real basis for the sad divisions and schisms that subsequently arose between us concerning the doctrine of Incarnation. In words and life we confess the true doctrine concerning Christ our Lord, notwithstanding the differences in interpretation of such a doctrine which arose at the time of the Council of Chalcedon. ”
According to the canons of the Oriental Orthodox Churches, the four Archbishops of Rome, Alexandria, Ephesus (later transferred to Constantinople) and Antioch were all given status as Patriarchs, or in other words, the Ancient Apostolic Centers of Christianity by the First Council of Nicea (predating the schism) — each of the four being responsible for those bishops and churches under his jurisdiction within his own quarter of Christendom, being the Metropolitan Archbishop of the Province, (with the exception of the Archbishop or Patriarch of Jerusalem, who was to be independent of all of these.) Thus, the Archbishop of Rome (ie, the Pope of the Catholic Church) has always been held by the others to be in Communion, and fully sovereign within his own quadrant.
The technical reason for the schism was that the Bishop of Rome excommunicated the non-Chalcedonian bishops in 451 AD for refusing to accept the "in two natures" teaching, thus declaring them to be out of communion with him, although they have continued to recognize him as an equal.
(Wikipedia, "Oriental Orthodoxy")
The problem comes with the interpretation of that. Orthodox (and many Anglicans) hold that the primacy was more along the lines of "foremost of equals" (like a prime minister in a parliamentary system of government). catholic dogma teaches that the pope has headship or supremacy, and universal jurisdiction.
We contend that the Orthodox broke from us because (we would argue) we continued the unbroken tradition of what came before (including the papacy). I've maintained, among other things, that we still have ecumenical councils (and at the Councils of Florence [15th c.] and Lyons , we even invited the Orthodox and almost achieved a reunion). In two papers on this question (one / two), I showed how, e.g., the east had split off from the west on five different occasions; sometimes for several dozen years. In all five cases, the west was correct and orthodox, from both eastern and western perspectives today.
So, then, I contended that 1054 was simply yet another instance of this schismatic tendency of the east, where they were wrong once again. Once I had an Orthodox priest give a guest talk in a discussion group at my home and when I asked him about this very thing, he (literally) just shrugged his shoulders and couldn't respond to it. No one has, since I've made this argument.
Of course (as we would expect) the Orthodox think we departed from them, and that they maintained the mainstream apostolic tradition. They argue that papal power had become too great and that the filioque clause was a corruption and illegitimate addition to the Nicene Creed, whereas we say it was a consistent development of trinitarian theology, rightly understood.
Mere politics and cultural differences are always factors in these things, as well as the differences in language. The east was subject to the strong tendency of caesaropapism (making the emperor in effect or in actuality the head of the Church, with the state being over the Church): precisely the error that the papacy allows the Church to largely avoid.
As for the question of apostolic succession; in the case of Orthodoxy, we acknowledge their possession of the succession, because they had validly ordained bishops from the previous age when east and west were united. They continued that, so that all their ordinations are valid and apostolic, as we recognize. As for Anglicans, they claim the same thing on the same basis, but we argue that they changed the ordination rites in the 16th century, thus bringing about an invalid ordination, and hence, loss of apostolicity. There can be some exceptions in some cases . . .
Various issues contributed to the schism. I've outlined them in my introductory paper. Orthodox tend to see the west as over-rational and insufficiently mystical. I say this is a caricature, but it is true that we place relatively more emphasis on reason than they do. We think our view and approach to the faith is balanced and multi-faceted, but they think it is too far in one direction. They also think we are overly-dogmatic, and should not have defined many things that we did (such as transubstantiation); that we require things that should be left to individual opinion.
As for inter-communion: many Orthodox jurisdictions are more opposed to us than we are to them. If they had been more open to the ecumenical process, the reunion would already have been accomplished by now (or in the 15th or 13th centuries). There has generally been a great "anti-Latin" animus, that goes back to the Crusades and the sacking of Constantinople in 1204, etc. (that was not approved by the pope at all).