By "the third decade" I mean books 21-30 of Livy's 142-book history of Rome. The first decade contains books 1-10, which cover the history of Rome from its mythical beginnings up until the early third century BC and the late stages of the Samnite Wars; the second decade, books 11 through 20, continue the story up until just before the Second Punic War. Books 11-20 are missing, except for a scrap of papyrus containing 41 words of book 11, found by archaeologists in Egypt in the 1980's.
The third decade, books 21-30, covers the Second Punic War, Rome's war against Carthage under its brilliant leader Hannibal, who actually came pretty close to conquering Rome, occupying much of Italy for well over 10 years before ha was finally defeated, hunted down and killed and Rome defeated Carthage for the 2nd of 3 times. (The 3rd time, Rome destroyed Carthage.)
It's debated these days just how good an historian Livy really is, how reliable the historical information is we get from him, how careful he is to get all the facts right. But even some of those who most emphatically denigrate him as an historian still praise him highly as an author. Apart from how accurate the tales he tells are as history, they are dramatic, exciting, gripping tales well-told. Livy is a great read. And of the surviving parts of his history, Livy's third decade seems to be the most popular, considered to be the most exciting reading. I would say that it's up there with ancient Rome's most renowned verse. Michael Reeve, a professor of Classics at Cambridge, said, at a colloquium in 1987, of a passage from Livy's book 23, that it "makes me wonder why our pupils spend so much of their time reading verse." ( Studies in Latin literature and its tradition: In honour of C.O. Brink,pp 103-104. )
The third decade seems to have been one of the best-loved parts of Livy's history right from the start, which may have everything to do with why we still have a lot of manuscripts of it. (154 manuscripts of the third decade, according to Reeve, p 107, but that was in the late 1980's, the total may be higher now.) I've come across 2 web pages, each dedicated to one of those manuscripts: this 15th-century manuscript in the University of Glasgow may not be the most important one from the point of view of preserving our closest guesses as to Livy's original text, but it's very nice to look at, with illustrations like this:
Then there's the Codex Puteanus of Livy's second decade, perhaps the single most important manuscript for transmitting the text. (There are many famous Codices Puteani, all named after their former owner Claude Dupuy, so when discussing this one make sure everyone understands that you're referring to the 5th-century Codex Puteanus of Livy's second decade.) This link leads to a page on the website of the Bibliothèque nationale de France where you can click through high-quality color pictures of all 900+ pages of the Codex Puteanus. You have to click for a while to get to the interesting stuff, but don't worry, the actual manuscript really is in there. It starts on Screen 11. That's a 5th-century manuscript, folks. Pay no attention to the BnF website telling you otherwise. The BnF has a tremendous amount of wonderful manuscripts, and they have excellent librarians working there too. It's a magnificent place. But the people who make their website are unfortunately still, in the year Two Thousand and For Crying Out Loud, not geniuses. You have no idea what I went through to get you that link showing you the Codex Puteanus. That codex should be right up on the library's homepage or very near it. Anyway... I found it, eventually, but only because I AM a genius, and wanted very badly to show it to you. Enjoy.