We present a minimal model for the formation and migration of aeolian sand dunes in unidirectional winds. It combines a perturbative description of the turbulent wind velocity field above the dune with a continuum saltation model that allows for saturation transients in the sand flux. The latter are shown to provide a characteristic length scale, called saturation length, which is distinct from the saltation length of the grains. The model admits two different classes of solutions for the steady-state profile along the wind direction: smooth heaps and dunes with slip face. We clarify the origin of the characteristic properties of these solutions and analyze their scaling behavior. We also investigate in some detail the dynamic evolution of heaps and dunes, including the steady-state migration velocity and transient shape relaxation. Although the minimal model employs nonlocal expressions for the wind shear stress as well as for the sand flux, it is simple enough to serve as a very efficient tool for analytical and numerical investigations and opens up the way to simulations of large scale desert topographies.