Recent neural network architectures such as the basic recurrent neural network (RNN) and Gated Recurrent Unit (GRU) have gained prominence as end-to-end learning architectures for natural language processing tasks. But what is the computational power of such systems? We prove that finite precision RNNs with one hidden layer and ReLU activation and finite precision GRUs are exactly as computationally powerful as deterministic finite automata. Allowing arbitrary precision, we prove that RNNs with one hidden layer and ReLU activation are at least as computationally powerful as pushdown automata. If we also allow infinite precision, infinite edge weights, and nonlinear output activation functions, we prove that GRUs are at least as computationally powerful as pushdown automata. All results are shown constructively.