The Septuagint manuscripts we have are in many cases bound up with the same manuscripts that serve as witnesses for the Greek New Testament. Examples:
- Codex Alexandrinus, c. 450
- Codex Vaticanus, c. 350
- Codex Ephraemi, c. 450
I don't know if I can improve on the other excellent answer to this question, other than to address the issue of what version of the Septuagint GREEKS (at least Greek Orthodox Christians) actually consider "official".
There are two versions in use:
A version published by the Apostoliki Diakonia in Greece by the Greek Orthodox Church. It is available online here. It is based on the Rahlf's edition, with some corrections made to reflect how some verses were quoted in antiquity (c. 80-400 or so) by Church Fathers.
An 1821 version that was originally published in Moscow, based on the Codex Alexandrinus. It is not available online. It might be available from some Greek publisher, but I don't know which.
A discussion on one Orthodox forum provided the following background:
The Orthodox LXX text is Published currently by two organizations. The First being the Apostoliki Diakonia with is the publishing and evangilation arm of the Greek Orthodox Church of Greece. The Second is the ZOE BROTHERHOOD which although their members are pious Greek Orthodox Christians and members of the Church of Greece are NOT affiliated with them. Both text, minus the footnotes, are exactly the same. The footnotes are 95% the same with the ZOE BROTHERHOOD LXX text, which still has the better notes.
The LXX text used by the Church of Greece is Alfred Rahlf's Septuaginta text made to conform as much as possible to traditional Orthodox renderings of the LXX as found in the writings of the Fathers and the Lucianic LXX text tradition, which was highly favored by many important Church Fathers such as St. John Chrystostom and others from Antioch and its shere of influence, as well as the liturgical readings of the LXX in the Church.
The LXX text was modified by Archmandrite Vambas with latter modifications and revisions done by professors and clergy from the University of Athens and Thessaloniki. Thus, this is the Modified-Rahlf's Septuaginta text is the Official LXX text of the Church of Greece and the Patriarch of Constandinople. However, only the Church of Greece has officially endorsed it by placing a Holy Synodial seal on the text. The Patriarchate has only put an official seal/endorsement on the New Testament Text. This makes sense as the LXX was corrected, modified and revised in Greece and the NT was modified, corrected and revised in Constantinople.
Mt. Athos, an autonomous pan-Orthodox monastic center within Greece, falls under the Patriarchate of Constantinople and not the autonomous Church of Greece, so its practices are a little different:
The Church of Greece's LXX text is NOT the LXX text used most often on the Holy Mountain [Mt. Athos]. The Moscow edition of 1821 is the prefered LXX text which is mostly a re-print of Codex Alexandrinus. The Church of Greece LXX text is the better LXX text, but the Monks received this text sortly after the Ottoman yoke was lifted and have used it ever since. The monks are very conservative to say the least. This is not to say that many monks on Mt. Athos don't use the Synod LXX text, because they do, its just that the majority of monks do not. In any event for your purposes it does not matter as the Moscow LXX text is nowhere to be found on the net.