In which other churches do Roman Catholics recognize apostolic succession? The Roman Catholic Church derives much of its claim to legitimacy from the teaching of "apostolic succession". Other churches also believe that their priests derive their authority in an unbroken chain of succession. I believe that the Roman Catholic Church acknowledges this of some other churches, accepting that the Eastern Orthodox priests and bishops have apostolic succession. Is this true? And which other churches would have their claims to apostolic succession recognized by the Roman Catholic Church?
I am not really satisfied with any of the answers given here, because they do not seem to answer the question.

First of all, apostolic succession refers to the fact that all bishops (that is, all those who have received the fullness of Holy Orders) can trace their Holy Orders all the way back to the Apostles. (See Catechism of the Catholic Church [CCC] 77-79. I will observe that the documentation detailing episcopal consecration only goes back to the Renaissance; however the Catholic Church holds that today’s Holy Orders form an unbroken line to the Apostles.)

Therefore, apostolic succession refers neither to doctrinal orthodoxy, nor to communion with the Bishop of Rome: it refers to the validity of holy orders in a given Church.

In addition to its own Holy Orders (both in the Western and the sui-iuris Eastern Catholic Churches), the Catholic Church recognizes the validity of the Holy Orders (hence of apostolic succession) in the following churches:


All of the Chalcedonian Orthodox Churches, also known as the Eastern Orthodox Church.
All of the non-Chalcedonian Oriental Orthodox Churches, such as the Coptic, Ethiopian, and Eretrian Orthodox Churches, as well as the Armenian Apostolic Church).
The non-Ephesian Churches (such as the Assyrian Church of the East).
The Polish National Catholic Church.


(Source: Wikipedia article “Holy Orders (Catholic Church).”) 

Notably absent from this list are the members of the Anglican Communion. As Pope Leo XIII taught in his apostolic letter Apostolicae Curae, the Catholic Church holds that the Anglican Church lost its ability to consecrate bishops in the 16th Century, when it changed its rite for episcopal consecration. (Some individual bishops may have found ways to be ordained validly, but it cannot be affirmed with certainty as a body.)

It is possible that other smaller groups, such as the Old Catholic churches, maintain valid Holy Orders, but there is no official recognition of such groups by the Holy See.