Ratings board listings might have stolen the thunder about the existence of definitive editions of the Mafia games, publisher 2K still had one nice surprise in the games’ official announcement: Mafia II: Definitive Edition and Mafia III: Definitive Edition are available right now, on PlayStation 4, Xbox One, and Steam.
Mafia II: Definitive Edition is a “lovingly remastered” version of Mafia II, which first came out in 2010, with improved visuals and audio and all DLC included. Mafia III is only a few years old, so its Definitive Edition isn’t a substantial change from the original, though it does include all DLC.
The original, un-remastered version of Mafia II remains intact on Steam as Mafia II (Classic); it’s not being sold as a standalone anymore, but its included with purchases of Mafia II: Definitive Edition. Furthermore, anyone who already owns Mafia II on Steam will get the Definitive Edition for free. Likewise, anyone who owns Mafia III on Steam, PlayStation 4, or Xbox One will be upgraded to its definitive edition.
2K is also working on a Definitive Edition of the original Mafia, though that won’t be released until August 28th. Rather than a remaster, Mafia: Definitive Edition is a being rebuilt from the ground up—the original being almost 20 years old now. It’s billed as “a faithful but expanded 4K- and HDR-ready remake that not only makes dramatic improvements to the original’s visuals and gameplay, but also builds upon its award-winning narrative and adds new features like motorcycles and collectibles”.

Finally, 2K have also released Mafia: Trilogy, a bundle that includes all three games (though Mafia: Definitive Edition obviously won’t be available until it’s released). If you don’t already own any of the versions eligible for a free upgrade, this works out to be a much cheaper way to get the full collection.
The Mafia games are all open-world crime dramas in the vein of Grand Theft Auto, but with a more serious tone; the series storytelling, in particular, has been widely praised. Each game takes place in a different era: the 1930s for Mafia, the 1940s and early 1950s for Mafia II, and the 1960s for Mafia III.