This chapter gives a thorough examination of kernel functions \(k(\mathbf {x}, \mathbf {x}')\) which are the primary driver of a GP model. We review the most common examples of stationary and non-stationary kernel families, discuss kernel composition and survey kernel selection approaches. The latter half of the chapter considers convergence and universal approximation properties of GP surrogates as the training set grows and then reviews links between GPs and stochastic differential equations. The Chapter is accompanied by a Python Jupyter notebook illustrating fitting of different GP kernels and prior mean functions to a synthetic one-dimensional dataset.