What I mean by the title is that a game series achieves significant sales in one gaming generation compared to its installments in the previous generation (even if not the series's first game that generation). I figure this is a good place to discuss when this happens and why.
Now some series hit it big right off the bat (Super Mario, Gran Turismo, and Halo, just to name one from each first party*), and some even break through in the same generation the series starts (Dragon Quest). But some just have low to good sales until a generation or so later when the sales take off.
Also, there wouldn't be a hard sales line to count, but 2-3 million more at least would be a good baseline.
Anyway, let's look at some of the cross-generation breakthroughs and why they did so well. In parentheses will be the breakthrough game's sales (in millions), followed by the highest selling game before that.
Perhaps the three most iconic of this phenomenon are Street Fighter II (can't really count the quarters, but it's safe to say it was a hell of a lot more than I, even from the first version alone), Final Fantasy VII (9.72 - Final Fantasy VI: 3.42), and Grand Theft Auto III (11.60 - GTA 2: 3.42). Okay, the first is a little gray, since arcades were 16-bit long before home consoles, but the first game came out in 1987, and II was in 1991.
Of course there are plenty of others (which I encourage you to bring up in the replies), but one odd one is Mario Kart. The series had actually broken through already, with MK 64 selling just under ten million copies, but then MK DS has nearly doubled that, and MK Wii had gone well over double that. One could argue userbase growth, but that only goes so far, since breakthroughs have happened on other systems with smaller userbases. There has to be something more.
* Then Pokemon, Zelda, Super Smash Bros, Resident Evil, Doom, Half-Life, Gears of War, Assassin's Creed, God of War, Crash Bandicoot (while still under Naughty Dog), Spyro the Dragon (while still under Insomniac), Donkey Kong, Donkey Kong Country, Pac-Man, Space Invaders, Double Dragon, Final Fight, Starcraft, Diablo...
A flashy-first game is awesome when it comes out. A great-first game is awesome forever.
Plus, just for the hell of it: Kelly Brook at the 2008 BAFTAs




















