Other villains still roam the streets, so you might turn a corner and get into a scuffle with another bad guy. However, they might cut you in on their action, too - if you defeat a small-time hood, you might inherit their mission, spawning another side quest. Now in addition to robbing a bank, you'll need to set a building on fire or steal gear from a pawn shop. Unfortunately for you, the pawn shops have security cameras, so you'll need to sneak around a bit to avoid detection. Pull off the side jobs, though, and you'll be rewarded with even more extra time.
Assuming you finally make it to the vault and can grab the cash, there's one final hurdle you'll need to face: a heavy-duty hero. Signature supers like Manticore, Positron, and even Statesman himself may appear to stop you from getting away; the good guys are randomly chosen for each mission. These won't be easy fights; these will be epic brawls that make you glad you fight dirty.
You'll want to bring friends for the spectacle of Mayhem Missions, but if you're a solo player, the developers say you can handle them on your own, since content will scale to the number of players. Look for Mayhem Missions as part of the Issue 7 update to City of Villains, which will be delivered free of charge to subscribers in late April or early May.
