We consider sequential decision problems under uncertainty, where we seek to optimize an unknown function from noisy samples. This requires balancing exploration (learning about the objective) and exploitation (localizing the maximum), a problem well-studied in the multi-armed bandit literature. In many applications, however, we require that the sampled function values exceed some prespecified “safety” threshold, a requirement that existing algorithms fail to meet. Examples include medical applications where patient comfort must be guaranteed, recommender systems aiming to avoid user dissatisfaction, and robotic control, where one seeks to avoid controls causing physical harm to the platform. We tackle this novel, yet rich, set of problems under the assumption that the unknown function satisfies regularity conditions expressed via a Gaussian process prior. We develop an efficient algorithm called SafeOpt, and theoretically guarantee its convergence to a natural notion of optimum reachable under safety constraints. We evaluate SafeOpt on synthetic data, as well as two real applications: movie recommendation, and therapeutic spinal cord stimulation.