In recent years, there has been extensive study of the mathematical basis of weather prediction leading to a new system of continuous equations that are well posed, and a set of conditions that make discrete atmospheric and other models stable and potentially more accurate. In particular, the theory deals with initial boundary value problems that admit multiple timescales. Using this theory, a quasi-nonhydrostatic model called QNH was developed at NOAA’s Forecast Systems Laboratory. The model is fully compressible and explicit in the vertical as well as the horizontal direction. It is characterized by a parameter, ‘‘a’’ (typically the square of the vertical to horizontal aspect ratio), which multiplies the hydrostatic terms in the vertical equation of motion. In this paper, the authors describe the theoretical basis for the use of these models in mesoscale weather prediction. It is shown that for the mesoscale, the parameter has the effect of decreasing both the frequency and amplitude of the gravity wave perturbation response to small-scale impulses in forcing and to unbalanced initial conditions. This allows a modeler to choose a length scale below which gravity wave generation is suppressed. A weakness of the approach is that the hydrostatic adjustment process is slowed down. The analysis indicates that the parameter does not have an effect on the Rossby waves, the larger horizontal-scale gravity waves, nor on forced solutions such as those created by heating. The bounded derivative initialization is discussed. Since the speeds of the vertical acoustic waves are decreased, quasi-nonhydrostatic models can calculate the vertical equations explicitly and still meet the Courant–Friedrichs–Levy criteria. It is concluded that the unique characteristics of quasi-nonhydrostatic models may make them valuable in mesoscale weather prediction, particularly of clouds and precipitation.