Flexoelectricity refers to the linear coupling of strain gradient and electric polarization. Early studies of this subject mostly look at liquid crystals and biomembranes. Recently, the advent of nanotechnology revealed its importance also in solid structures, such as flexible electronics, thin films, energy harvesters, etc. The energy storage function of a flexoelectric solid depends not only on polarization and strain, but also strain-gradient. This is our basis to formulate a consistent model of flexoelectric solids under small deformation. We derive a higher-order Navier equation for linear isotropic flexoelectric materials which resembles that of Mindlin in gradient elasticity. Closed-form solutions can be obtained for problems such as beam bending, pressurized tube, etc. Flexoelectric coupling can be enhanced in the vicinity of defects due to strong gradients and decay away in far field. We quantify this expectation by computing elastic and electric fields near different types of defects in flexoelectric solids. For point defects, we recover some well-known results of non-local theories. For dislocations, we make connections with experimental results on NaCl, ice, etc. For cracks, we perform a crack-tip asymptotic analysis and the results share features from gradient elasticity and piezoelectricity. We compute the J integral and use it for determining fracture criteria. Conventional finite element methods formulated solely on displacement are inadequate to treat flexoelectric solids due to higher order governing equations. Therefore, we introduce a mixed formulation which uses displacement and displacement-gradient as separate variables. Their known relation is constrained in a weighted integral sense. We derive a variational formulation for boundary value problems for piezeo- and/or flexoelectric solids. We validate this computational framework against exact solutions. With this method more complex problems, including a plate with an elliptical hole, stationary cracks, as well as structures with periodic structures, can be studied consistently with the continuum theory. We also generate predictions of experimental merit and reveal interesting flexoelectric phenomena with potential for application.