The linear theory of internal gravity waves and the nonlinear development of undular bores supported by mesospheric thermal inversion layers is presented. Whenever the buoyancy frequency has a local maximum as a result of a local increase in the potential temperature gradient, internal waves having a frequency between the maximum and minimum values of the buoyancy frequency can exist. Two-dimensional nonlinear solutions of a simplified model show that long-wavelength initial disturbances of finite amplitude, nonlinearly steepen, and form undulatory bores and that the analysis and numerical solutions presented show that mesospheric bores are consistent with nonlinear internal gravity waves trapped within a thermal inversion layer. The implications of linear theory and numerical solutions to the observations of mesospheric bores are discussed. In particular, the simulations agree with the observed separation of wave crests, which is in consistent with the notion that a mesospheric bore is composed of a train of solitary waves each with an amplitude-dependent speed.