In the space of polynomial automorphisms of C 2 , the set of structurally stable maps is not dense. To obtain this result we develop some of the theory of moduli of stability for holomorphic maps. Also, the set of hyperbolic maps is not dense in the set of polynomial automorphisms; however, given any polynomial automorphism, F , of C 2 , there is a polynomial automorphism, G, of the same degree as and arbitrarily near F such that each periodic point of G is hyperbolic.