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We study ground states of a nonlinear Schr\"odinger equation driven by the infinitesimal generator of a rotationally invariant Levy process. The equation includes many special cases such as classical Schrodinger equations, fractional Schrodinger equations and relativistic Schrodinger equations, etc. It is proved that the equation possesses ground states in a suitable space of functions, then the regularity of solutions to the equation is examined, in particular, any solution is H\"older continuous, and, if the process involves diffusion terms, any solution is twice differentiable further. Finally, we show that any ground state is either positive or negative.