Intermolecular forces are pervasive in nature and give rise to various phenomena including surface wetting1, adhesive forces in biology2,3, and the Casimir effect4, which causes two charge-neutral, metal objects in vacuum to attract each other. These interactions are the result of quantum fluctuations of electromagnetic waves and the boundary conditions imposed by the interacting materials. When the materials are optically anisotropic, different polarizations of light experience different refractive indices and a torque is expected to occur that causes the materials to rotate to a position of minimum energy5,6. Although predicted more than four decades ago, the small magnitude of the Casimir torque has so far prevented direct measurements of it. Here we experimentally measure the Casimir torque between two optically anisotropic materials-a solid birefringent crystal (calcite, lithium niobite, rutile or yttrium vanadate) and a liquid crystal (5CB). We control the sign and strength of the torque, and its dependence on the rotation angle and the separation distance between the materials, through the choice of materials. The values that we measure agree with calculations, verifying the long-standing prediction that a mechanical torque induced by quantum fluctuations can exist between two separated objects. These results open the door to using the Casimir torque as a micro- or nanoscale actuation mechanism, which would be relevant for a range of technologies, including microelectromechanical systems and liquid crystals.