Deep neural networks implement a sequence of layer-by-layer operations that are each relatively easy to understand, but the resulting overall computation is generally difficult to understand. An intuitive hypothesis is that the role of each layer is to reformat information to reduce the "distance" to the desired outputs. With this spatial analogy, the layer-wise computation implemented by a deep neural network can be viewed as a path along a high-dimensional manifold of neural representations. With this framework, each hidden layer transforms its inputs by taking a step of a particular size and direction along the manifold, ideally moving towards the desired network outputs. We formalize this intuitive idea by leveraging recent advances in metric representational similarity. We extend existing representational distance methods by defining and characterizing the manifold that neural representations live on, allowing us to calculate quantities like the shortest path or tangent direction separating representations between hidden layers of a network or across different networks. We then demonstrate these tools by visualizing and comparing the paths taken by a collection of trained neural networks with a variety of architectures, finding systematic relationships between model depth and width, and properties of their paths.