We introduce a practical way of introducing convolutional structure into Gaussian processes, which makes them better suited to high-dimensional inputs like images than existing kernels. The main contribution of our work is the construction of an inter-domain inducing point approximation that is well-tailored to the convolutional kernel. This allows us to gain the generalisation benefit of a convolutional kernel, together with fast but accurate posterior inference. We investigate several variations of the convolutional kernel, and apply it to MNIST and CIFAR-10 that have been known to be challenging for Gaussian processes. We also show how the marginal likelihood can be used to find an optimal weighting between convolutional and RBF kernels to further improve performance. We hope this illustration of the usefulness of a marginal likelihood will help to automate discovering architectures in larger models.