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A game where you have to start over from the beginning when you die. Often goes together with procedural generation, so it doesn't get boring!

Near infinitely replayable games that usually feature, random generation, perma-death, and brutal difficulty. 

Shoot 'Em Up or shmup games are a subset of shooter games where the player continually moves forward while waves of enemies enter the screen. Both shooting and dodging are critical to advancing. See also Bullet Hell.