For any group G and set A, let CA(G;A) be the monoid of all cellular automata over the configuration space AG. In this talk, we present some algebraic results on CA(G;A) when G and A are both finite. First, we show that any generating set of CA(G;A) must have a cellular automaton with minimal memory set equal to G itself. Second, we describe the structure of the group of units of CA(G;A) in terms of a set of representatives of the conjugacy classes of subgroups of G. Third, we discuss the minimal cardinality of a generating set of CA(G;A): in some cases we give it precisely, while in others we give some bounds. We apply this to provide a simple proof that CA(G;A) is not finitely generated for various kinds of infinite groups G.