The QCD description of exclusive decays of the $\Lambda_b$ baryon involves hadronic matrix elements of non-local light ray operators, the light-cone distribution amplitudes. We introduce the complete set of three--quark distribution amplitudes and calculate the renormalization scale dependence for the leading twist. At leading order in the strong coupling the evolution is driven by pairwise two--quark interactions: heavy--light involving Sudakov logarithms as in the B-meson case, and light--light as in light mesons. We solve the evolution equation and show that its main effect is to generate a radiative tail extending to high energies. Finally, we present simple models for the distribution amplitudes based on QCD sum rules, and study the effect of the evolution on these models.