We define a notion of quantum or non-commutative, ergodicity for a class ofC *-dynamical systems (A, G, α) which we callquantized GNS systems. Such a system possesses a natural classical limit state ω, which induces a classical limit system by the GNS construction. The criterion for quantum ergodicity is that the time average 〈A〉 of an observable A ∈A equals the “space average” ω(A)I plus an errorK which is negligible in the classical limit. We prove that ergodicity of ω is a sufficient condition for quantum ergodicity of (A, G, α) if the classical limit system is abelian, give a conditional converse, and discuss a number of applications.