A general formulation for the study of electrostatic oscillations and streaming instabilities in bounded and unbounded Vlasov plasmas with transverse density gradients in a strong axial magnetic field is presented. It is found that the behavior of the plasma depends on a parameter which is a function of geometry, scale of inhomogeneity, and wavelength. Using this formulation, several experimentally realistic configurations have been studied. Two results obtained are of special interest. They are (1) for waves with the same wavelength, a density inhomogeneity and/or boundary increase Landau damping. (2) Under appropriate conditions, depending on the geometry and the degree of inhomogeneity, it is not possible to excite the two‐stream instability for any drift velocity. Previous theories of uniform plasmas appear as special cases in this general treatment.