Bootstrap percolation (BP) models are systems where sites are initially randomly occupied. Those sites that do not maintain a suitable local environment of occupied sites are successively removed. This culling process can be identified with a cellular automation. Variations of the local rules concerning suitable environments, lead to different families of models, which have members with percolation transitions of the usual type, as well as representatives undergo different types of first order transitions. The latter transitions are manifestations of longer ranged phenomena in the systems. In this review different families of the BP type are introduced and their relationships summarized. The finite-size effects observed near the first order transitions are also discussed. Recent exact and numerical results are presented, and some applications and open problems are outlined.