q-Levy processes are a new family of non-commutative random processes, consisting of operators represented on a q-deformed full Fock space. They provide an interpolation between the usual processes with independent increments, and the processes with freely independent increments appearing in free probability. I will describe a number of aspects of these processes. From a probabilistic perspective, I will show that all of them have a generalized chaos decomposition property. This in turn will lead to some preliminary results on the von Neumann algebras they generate.