Flexoelectricity possesses two gradient-dependent electromechanical coupling effects: the direct flexoelectric effect and the converse flexoelectric effect. The former can be used for sensing and energy generation; the latter can be used for ultraprecision actuation and control applications. Due to the direct flexoelectricity and large deformations, theoretical fundamentals of a generic nonlinear distributed flexoelectric double-curvature shell energy harvester are proposed and evaluated in this study. The generic flexoelectric shell energy harvester is made of an elastic double-curvature shell laminated with flexoelectric patches and the shell experiences large oscillations, such that the von Karman geometric nonlinearity occurs. Flexoelectric output voltages and energies across a resistive load are evaluated using the current model in the closed-circuit condition when the shell is subjected to harmonic excitations and its steady-state voltage and power outputs are also calculated. The generic flexoelectric shell energy harvesting theory can be simplified to shell (e.g., cylindrical, conical, spherical, paraboloidal, etc.) and nonshell (beam, plate, ring, arch, etc.) distributed harvesters and the simplification procedures are demonstrated in three cases, i.e., a cylindrical shell, a circular ring and a beam harvester. Other shell and nonshell flexoelectric energy harvesters with standard geometries can also be defined using their distinct two Lamé parameters and two curvature radii.