Abstract Our main result is a theorem that gives, in a certain setting, a necessary and sufficient condition under which multidimensional shift-invariant input-output maps with vector-valued inputs drawn from a certain large set can be uniformly approximated arbitrarily well using a structure consisting of a linear preprocessing stage followed by a memoryless nonlinear network. The inputs considered need not be continuous, and noncausal as well as causal maps are addressed. Approximations for noncausal maps for which inputs and outputs are functions of more than one variable are of current interest in connection with, for example, image processing.