We present two new methods for inference in Gaussian process (GP) models with general nonlinear likelihoods. Inference is based on a variational framework where a Gaussian posterior is assumed and the likelihood is linearized about the variational posterior mean using either a Taylor series expansion or statistical linearization. We show that the parameter updates obtained by these algorithms are equivalent to the state update equations in the iterative extended and unscented Kalman filters respectively, hence we refer to our algorithms as extended and unscented GPs. The unscented GP treats the likelihood as a 'black-box' by not requiring its derivative for inference, so it also applies to non-differentiable likelihood models. We evaluate the performance of our algorithms on a number of synthetic inversion problems and a binary classification dataset.