A general analytic theory for the distribution function of particles backstreaming from an arbitrary shock in a magnetized plasma (neglecting wave-particle scattering) is presented. The theory explicitly shows that abrupt cutoffs in the parallel velocity, where ∂f/∂‖v∥‖> 0, arise naturally in the upstream region, irrespective of the acceleration processes active at the shock. These ''cutoff'' distributions may provide free energy for wave growth. Two classes of cutoff arise, escape cutoffs due to the particles requiring a characteristic speed to outrun the shock, and edge cutoffs due to the shock having an edge. Analytic expressions for the particle distribution function upstream of finite planar shocks and shocks whose two-dimensional sections are parabolic are derived. A theory for the electron distribution function upstream from the earth's bow shock and the source of free energy for the observed Langmuir waves is presented. The electron distribution has an escape cutoff in the parallel velocity. This cutoff distribution is the source of free energy for Langmuir wave growth. Distribution functions and the spatial variation of the cutoff velocity are illustrated graphically. The theory is consistent with the spatial variation in the electron beam velocity inferred from indirect observations, assuming the observed electron beams to be the wave-particle scattered remnants of cutoff distributions. An application of this theory to type II solar radio bursts is sketched. ¿American Geophysical Union 1987