We will first set up the basic terminology of coupled cell networks. We then describe some very recent work about when two networks with different topologies or architectures can be dynamically equivalent. The aim of the talk will be to present the main results and idea of the proofs. The proofs are algorithmic in nature and give a systematic way to construct the desired equivalence. These results give a complete treatment for the case of continuous coupled dynamical systems. Some of the results we present are unexpected and may have applications in numerical analysis and the design of systems that are robust to failures of individual units. Joint NSF supported research with Dr. Mike Field.