Statistical depth functions are a class of functions that provide a center- outward ordering of sample points in multidimensional space. In this work we intro- duce a novel depth function that is based on the cumulative distribution function along random directions, and is thus termed directional distribution depth. Some properties and a connection to the Mahalanobis depth when applied to sphered data are shown. The proposed depth is used as a basis for supervised classification using maximum depth classifiers and more flexible polynomial separators in the depth space. It is shown to be effective and competitive against other depth functions through simulated experiments and real data applications.