The amount of power that a photovoltaic (PV) power plant generates depends on the DC voltage that is applied to the PV panels. The relationship between this control input and the generated power is non-convex and has multiple local maxima. Moreover, since the generated power depends on time-varying environmental conditions, such as solar irradiation, the location of the global maximum changes over time. Maximizing the amount of energy that is generated over time is known as the maximum power point tracking (MPPT) problem. Traditional approaches to solve the MPPT problem rely on heuristics and data-based gradient estimates. These methods typically converge to local optima and thus waste energy. Our approach formalizes the MPPT problem as a Bayesian optimization problem. This formalization admits algorithms that can find the maximum power point after only a few evaluations at different input voltages. Specifically, we model the power-voltage curve as a Gaussian process (GP) and use the predictive uncertainty information in this model to choose control inputs that are informative about the location of the maximum. We extend the basic approach by including operational constraints and making it computationally tractable so that the method can be used on real systems. We evaluate our method together with two standard baselines in experiments, which show that our approach outperforms both.