This paper introduces the notion of entropy dimension to measure the complexity of zero entropy dynamical systems, including the probabilistic and the topological versions. These notions are isomorphism invariants for measure-preserving transformation and continuity. We discuss basic propositions for entropy dimension and construct some examples to show that the topological entropy dimension attains any value between 0 and 1. This paper also gives a symbolic subspace to achieve zero topological entropy, but with full entropy dimension.
