In this paper, we analyze some classes of the abstract non-scalar Volterra difference equations depending of several variables. We introduce the notion of a discrete multi-dimensional $(k,C,B,(A_{i})_{1\leq i\leq n},({\rm v}_{i})_{1\leq i\leq n})$-existence family and the notion of a discrete $(k,C,B,(A_{i})_{1\leq i\leq m},({\rm v}_{i})_{1\leq i\leq m},{\mathcal I})$-existence family, providing also certain results about the existence and uniqueness of almost periodic type solutions to the abstract non-scalar Volterra difference equations of several variables.