In this paper, we determine analytical upper bounds on the local Lipschitz constants of feedforward neural networks with ReLU activation functions. We do so by deriving Lipschitz constants and bounds for ReLU, affine-ReLU, and max pooling functions, and combining the results to determine a network-wide bound. Our method uses several insights to obtain tight bounds, such as keeping track of the zero elements of each layer, and analyzing the composition of affine and ReLU functions. Furthermore, we employ a careful computational approach which allows us to apply our method to large networks such as AlexNet and VGG-16. We present several examples using different networks, which show how our local Lipschitz bounds are tighter than the global Lipschitz bounds. We also show how our method can be applied to provide adversarial bounds for classification networks. These results show that our method produces the largest known bounds on minimum adversarial perturbations for large networks such as AlexNet and VGG-16.