We provide in this paper a systematic development of nonlinear stochastic difference equations driven by martingales (that depend on a spatial parameter); three such equations are considered. We begin with the existence and uniqueness of solutions and continue with the study of stochastic properties, such as the martingale and Markov properties, along with ϕ irreducibility and recurrence. We discuss in the final section the discrete-time flow and asymptotic flow properties of the solution process.