As the share of renewables in the grid increases, the operation of power systems becomes more challenging. The present paper proposes a method to formulate and solve chance-contrained optimal power flow while explicitly considering the full nonlinear AC power flow equations and stochastic uncertainties. We use polynomial chaos equations to model the effects of arbitrary uncertainties of finite variance, which enables to predict and optimize the system state for a range of operating conditions.We apply chance constraints. Our method incorporates a more detailed and a more flexible description of both the controllable variables and the resulting system state than previous methods. Two case studies highlight the efficacy of the method, with a focus on satisfaction of the AC power flow equations and on the accurate computation of moments of all random variables.