Quantum entanglement(1,2) plays a vital role in many quantum-information and communication tasks(3). Entangled states of higher-dimensional systems are of great interest owing to the extended possibilities they provide. For example, they enable the realization of new types of quantum information scheme that can offer higher-information-density coding and greater resilience to errors than can be achieved with entangled two-dimensional systems (see ref. 4 and references therein). Closing the detection loophole in Bell test experiments is also more experimentally feasible when higher-dimensional entangled systems are used(5). We have measured previously untested correlations between two photons to experimentally demonstrate high-dimensional entangled states. We obtain violations of Bell-type inequalities generalized to d-dimensional systems(6) up to d = 12. Furthermore, the violations are strong enough to indicate genuine 11-dimensional entanglement. Our experiments use photons entangled in orbital angular momentum(7), generated through spontaneous parametric down-conversion(8,9), and manipulated using computer-controlled holograms.