Many landscapes appear scale-invariant, in the sense that it is difficult to judge their size without the help of a scale bar. But landscapes do contain characteristic scales. For example: it is possible to identify the smallest valleys in a landscape, and at finer scales the topography is essentially smooth; evenly spaced ridges and valleys create dominant topographic 'wavelengths'; and branching river networks are organized into a hierarchy of nested drainage basins. I present a framework for predicting these scales. In landscapes shaped by a combination of soil creep and stream incision, the long-term evolution of the topography can be described with a nonlinear advection-diffusion equation. The relative magnitudes of these two dominant erosion processes can therefore be expressed as a quantity analogous to a Peclet number. I use a numerical landscape evolution model to show that the aforementioned characteristic scales are functions of the Peclet number, and test this result by measuring the Peclet number for two landscapes in California and comparing the predicted scales with the observed topography. The favorable comparison implies that the characteristic scales observed in terrestrial and planetary drainage networks provide an easily observable record of long-term sediment transport and erosion rates.