  Motivated by homothetic solutions to curvature-driven flows of planar curves,
as well as their many physical applications, this work carries out a systematic
study of oriented curves whose curvature $\kappa$ is a given function of
position or direction. The analysis is informed by a dynamical systems point of
view. Though focussed on situations where the prescribed curvature depends only
on the distance $r$ from a distinguished point, the basic dynamical concepts
are seen to apply in other situations as well. As an application, a complete
classification of all simple closed solutions of $\kappa = ar^b$, with real
constants $a,b$, is established.
