Abstract <p>We investigate a nonlinear Schrödinger equation of general form in which the chromatic dispersion and the potential are given by two arbitrary functions. The equation is a natural generalization of a wide class of related nonlinear partial differential equations that are often encountered in various fields of theoretical physics and mechanics, including nonlinear optics and plasma physics. For several general nonlinear Schrödinger equations, new exact solutions in implicit form are found, which are expressed in terms of elementary and arbitrary functions. One-dimensional reductions are described, which reduce the considered partial differential equation to simpler ordinary differential equations or systems of such equations.</p>