Orbital stability theory for the localized bound states (solitons) of the Nonlinear Schr(o)dinger (NLS) equation is generalized to arbitrary potentials in any dimension and with a general-type nonlinear term.The key assumptions on the spectrum of the linearized NLS equation are identified and their physical meaning elucidated.This is corroborated by direct numerical simulations of multidimensional solitons with periodic and other lattice type potentials.