The stories usually follow a single group of survivors, caught up in the sudden rush of the crisis. The narrative generally progresses from the onset of the zombie plague, then initial attempts to seek the aid of authorities, the failure of those authorities, through to the sudden catastrophic collapse of all large-scale organization and the characters’ subsequent attempts to survive on their own.
Such stories are often squarely focused on the way their characters react to such an extreme catastrophe, and how their personalities are changed by the stress, often acting on more primal motivations (fear, self-preservation) than they would display in normal life.
The plot usually develops that internal personality clashes or politics leads to the group breaking apart, and the result is that the security of the compound is compromised, and the zombies get in.
Generally the zombies in these situations are the slow, lumbering and unintelligent kind first made popular in the 1968 film Night of the Living Dead. Recent films, however, have refeatured zombies that are more agile, vicious, intelligent, and stronger than the traditional zombie. In most cases of "fast" zombies, creators use mindless human beings (as in Zombieland and Left 4 Dead) instead of re-animated corpses to logically counter the "slow death walk" of Romero’s zombies.