Quantum computers are expected to offer substantial speedups over their classical counterparts and to solve problems that are intractable for classical computers. Beyond such practical significance, the concept of quantum computation opens up new fundamental questions, among them the issue whether or not quantum computations can be certified by entities that are inherently unable to compute the results themselves. Here we present the first experimental verification of quantum computations. We show, in theory and in experiment, how a verifier with minimal quantum resources can test a significantly more powerful quantum computer. The new verification protocol introduced in this work utilizes the framework of blind quantum computing and is independent of the experimental quantum-computation platform used. In our scheme, the verifier is only required to generate single qubits and transmit them to the quantum computer. We experimentally demonstrate this protocol using four photonic qubits and show how the verifier can test the computer's ability to perform measurement-based quantum computations.