We present the stochastic approach to nonlinear kinetic equations (without gradient terms) in a unifying general framework, which covers many interactions important in applications, such as coagulation, fragmentation, inelastic collisions, as well as source and efflux terms. We provide conditions for the existence of corresponding stochastic particle systems in the sense of regularity (nonexplosion) of a jump process with unbounded intensity. Using an appropriate space of measure-valued functions, we prove relative compactness of the sequence of processes and characterize the weak limits in terms of solutions to the nonlinear equation. As a particular application, we derive existence theorems for Smoluchowski's coagulation equation with fragmentation, efflux and source terms, and for the Boltzmann equation with inelastic collisions.