In this paper, we formulate Gaussian process regression with observations under the localization uncertainty. In our formulation, effects of observations, measurement noise, localization uncertainty and prior distributions are all correctly incorporated in the posterior predictive statistics. The analytically intractable posterior predictive statistics are proposed to be approximated by Laplace approximations in different degrees of complexity. Such approximations have been carefully tailored to our problems and their approximation errors and complexity are analyzed. Simulation results demonstrate that the proposed approaches perform much better than approaches without considering the localization uncertainty correctly.