We introduce a class of nonstationary covariance functions for Gaussian process (GP) regression. Nonstationary covariance functions allow the model to adapt to functions whose smoothness varies with the inputs. The class includes a nonstationary version of the Matérn stationary co- variance, in which the differentiability of the regression function is con- trolled by a parameter, freeing one from fixing the differentiability in advance. In experiments, the nonstationary GP regression model per- forms well when the input space is two or three dimensions, outperform- ing a neural network model and Bayesian free-knot spline models, and competitive with a Bayesian neural network, but is outperformed in one dimension by a state-of-the-art Bayesian free-knot spline model. The model readily generalizes to non-Gaussian data. Use of computational methods for speeding GP fitting may allow for implementation of the method on larger datasets.