There's much to like about this book. As a short biography of the great book hunter Bracciolini, it's extremely well done. The drive that he and other early humanists felt to rescue classical works from the ravages of time by preserving them in manuscript copies (and later in print) is plumbed to its depths here, and although there are many aspects of Bracciolini's discovery of Lucretius that we cannot know (he doesn't even say which monastic library he found it in), the framework Greenblatt has created does the job well.
Beyond Bracciolini, Greenblatt also takes us back to Lucretius' own day and the immediate aftermath of his work's composition; and slightly forward in time to the Villa of the Papyri at Herculaneum, where a wide variety of Epicurean treatises have been discovered among the charred scrolls. He explores the broad scope of early humanism (particularly that of the Bracciolini circle), and also delves into the lessons and principles espoused by Lucretius in his text.
I could have done with more on the later publishing history and reception of De rerum natura; there is some here, but even more on the Enlightenment-era views of the work would have been welcome (Jefferson had seven different editions of the work, for example). That said, Greenblatt's discussion of Montaigne's personal copy - now at Eton College - was a delight to read (and has, I find, been the subject of a book in its own right).
While I might quibble over the scope of the subtitle, I greatly enjoyed The Swerve, and recommend it.