We consider the entropy of systems of random transformations, where thetransformations are chosen from a set of generators of a ${\Bbb Z}^d $action.We show that the classical definition gives unsatisfactory entropy results inthe higher-dimensional case, i.e. when $d \geq 2$.We propose a definition of the entropy for random group actions whichagrees with the classical definition in the one-dimensional case, and whichgives satisfactory results in higher dimensions.This definition is based on the fibre entropy of a certain skew product.We identify the entropy by an explicit formula which makes it possible tocompute the entropy in certain cases.