In this paper we present a simple mathematical theory of carving turns in alpine skiing and snowboarding. The theory captures the basic dynamics of carving runs and thus provides a useful tool for assessing the potential and limitations of the carving technique. We also apply the model to simulate runs on slopes of constant gradient and describe the results. We find that pure carving is possible only on relatively flat slopes, with the critical slope angle in the range of 8-20 degrees. The exact value depends mostly on the coefficient of snow friction and to a lesser degree on the sidecut radius of the skis. In wiggly carving runs on slopes of subcritical gradient, the aerodynamic drag force remains relatively unimportant and the speed stops growing well below the one achieved in fall-line gliding. This is because the increased g-force of carving turns leads to enhanced snow friction which can significantly exceed the one found in the gliding. At the critical gradient, the g-force of carving turns becomes excessive. For such and even steeper slopes only hybrid turns, where at least some part of the turn is skidded, are possible. The carving turns of all alpine racing disciplines are approximately similar. This is because in the dimensionless equations of the model the sidecut radius of skis enters only via the coefficient of the aerodynamic drag term, which always remains relatively small. Simple modifications to the model are made to probe the roles of skier angulation and skidding at the transition phase of hybrid turns. As expected the angulation gives certain control over the turn trajectory, but does not remove the slope limitations.