This paper proposes novel reachability algorithms for both exact (sound and complete) and over-approximation (sound) analysis of deep neural networks (DNNs). The approach uses star sets as a symbolic representation of sets of states, which are known in short as stars and provide an effective representation of high-dimensional polytopes. Our star-based reachability algorithms can be applied to several problems in analyzing the robustness of machine learning methods, such as safety and robustness verification of DNNs. The star-based reachability algorithms are implemented in a software prototype called the neural network verification (NNV) tool that is publicly available for evaluation and comparison. Our experiments show that when verifying ACAS Xu neural networks on a multi-core platform, our exact reachability algorithm is on average about 19 times faster than Reluplex, a satisfiability modulo theory (SMT)-based approach. Furthermore, our approach can visualize the precise behavior of DNNs because the reachable states are computed in the method. Notably, in the case that a DNN violates a safety property, the exact reachability algorithm can construct a complete set of counterexamples. Our star-based over-approximate reachability algorithm is on average 118 times faster than Reluplex on the verification of properties for ACAS Xu networks, even without exploiting the parallelism that comes naturally in our method. Additionally, our over-approximate reachability is much less conservative than DeepZ and DeepPoly, recent approaches utilizing zonotopes and other abstract domains that fail to verify many properties of ACAS Xu networks due to their conservativeness. Moreover, our star-based over-approximate reachability algorithm obtains better robustness bounds in comparison with DeepZ and DeepPoly when verifying the robustness of image classification DNNs.