Abstract : The linearized theory of magneto-atmospheric waves (involving the combined restoring forces due to buoyancy, compressibility, and magnetic field) is developed for the case of a horizontal magnetic field which may vary with height. The convective stability of the basic atmosphere is considered, and it is shown that a nonuniform horizontal magnetic field may be destabilizing as well as stabilizing. A detailed model of a sunspot penumbra is studied in order to identify the mode of running penumbral waves. It is found that penumbral waves may be identified with magneto-atmospheric waves of the plus type that are vertically trapped at photospheric levels. An exact analytical solution for magneto-atmospheric wave modes is found in the case of an isothermal atmosphere permeated by a uniform horizontal magnetic field, without making the usual short-wavelength approximation. This solution is applied to an idealized model of the low-corona-chromosphere transition region as a model for flare-induced coronal waves. The exact solution mentioned above is also used in conjunction with a simple two-layer model of a sunspot penumbra to further study the mode of running penumbral waves. The lowest plus eigenmode of the model is in good agreement with observations of penumbral waves.