This survey concerns probabilistic models motivated by cooperative sequential adsorption (CSA) models. CSA models are widely used in physics and chemistry for modelling adsorption processes in which adsorption rates depend on the spatial configuration of already adsorbed particles. Corresponding probabilistic models describe random sequential allocation of particles either in a subset of Euclidean space, or at vertices of a graph. Depending on a technical setup these probabilistic models are stated in terms of spatial or integer-valued interacting birth-and-death processes. In this survey we consider several such models that have been studied in recent years.