Title: A theory of representation learning in neural networks gives a deep generalisation of kernel methods. Abstract: The successes of modern deep neural networks (DNNs) are founded on their ability to transform inputs across multiple layers to build good high-level representations. It is therefore critical to understand this process of representation learning. However, standard theoretical approaches (formally NNGPs) involving infinite width limits eliminate representation learning. We therefore develop a new infinite width limit, the representation learning limit, that exhibits representation learning mirroring that in finite-width networks, yet at the same time, remains extremely theoretically tractable. In particular, we derive an elegant objective that describes how each network layer learns representations that interpolate between input and output, and we confirm that the modes of the objective match the behaviour of finite but wide networks. Moreover, we use this limit and objective to develop a flexible, deep generalisation of kernel methods, that we call deep kernel machines (DKMs). We show that DKMs can be scaled to large datasets using inducing point methods from the Gaussian process literature, and we show that DKMs exhibit superior performance to other kernel-based approaches.