A material is said to be flexoelectric when it polarizes in response to strain gradients. The phenomenon is well known in liquid crystals and biomembranes but has received less attention in hard materials such as ceramics. Here we derive the governing equations for a flexoelectric solid under small deformation. We assume a linear constitutive relation and use it to prove a reciprocal theorem for flexoelectric materials as well as to obtain a higher-order Navier equation in the isotropic case. The Navier equation is similar to that in Mindlin's theory of strain-gradient elasticity. We also provide analytical solutions to several boundary value problems. We predict size-dependent electromechanical properties and flexoelectric modulation of material behavior. Our results can be used to interpret experiments on flexoelectric materials which are becoming increasingly sophisticated due to the advent of nanoscale probes.