It is first shown that, in the limit of infinitely short convective adjustment time, or strict Quasi Equilibrium (Emanuel, Neelin and Bretherton, 1994), the interface between moist (precipitating) and dry (non-precipitating) regions exhibits a discontinuity in the precipitation and vertical velocity fields. These discontinuities are refereed here to as precipitation fronts. They are not stationary, and can be classified into three categories according to the motion of the interface between the dry and moist regions. A drying front is associated with the dry region expanding into the moist regions. In this case, the front speed is intermediate between the propagation speed of a convectively coupled wave and that of an uncoupled wave. A slow moistening front corresponds to the interface moving into the dry regions with a speed that is lower than the speed of a convectively coupled wave. A fast moistening front occurs when the interface moves into the dry regions with a velocity that is higher than the speed of an uncoupled wave. The nature of the front and its velocity can be solely determined from the atmospheric properties on both side of the interface. The theoretical predictions for the strict quasi-equilibrium are tested in a numerical model, and are found to accurately predict the displacement of the interface between the dry and moist regions for realistic convective adjustment time. It is also argued that waves incident to a precipitation front are subject to partial reflection and transmission. This framework is applied to an idealized 2 dimensional Walker circulation on an equatorial beta-plane, for which it predicts the partial reflection and transmission of Rossby and Kelvin waves at the edge of the Walker circulation.