Quantum computers are among the most promising applications of quantum-enhanced technologies. Quantum effects such as superposition and entanglement enable computational speed-ups that are unattainable using classical computers. Blind quantum computing shows an additional fundamental advantage of quantum over classical computation: a computation can be made private. Using the framework of measurement-based quantum computing, blind quantum computing enables a nearly-classical client to access the resources of a more computationally-powerful quantum server without divulging the content of the requested computation. Here, we demonstrate the first experimental version of this protocol using polarization-entangled photonic qubits. We show various blind one- and two-qubit gate operations as well as blind versions of the Deutsch’s and Grover’s algorithms. Further, we address the question if quantum computations can be verified by classical entities which are not able to compute the results themselves. When the technology to build quantum computers becomes available, blind quantum computing and the verification of quantum computations will become an important feature of quantum information processing.