Gaussian processes provide both predictions at unseen data locations and associated quantification of uncertainty. It is difficult to select a Gaussian process model that correctly encodes some of the most important properties of the latent function. In this talk I discuss smoothness misspecification, which occurs when a Gaussian process model is either smoother or rougher than the truth. I review a number of recent results that demonstrate that smoothness misspecification need not be a problem when it comes to the rate of convergence of the posterior and the reliability of uncertainty quantification.