Over the past few years, important results have been obtained on time-varying feedback stabilization for strongly nonlinear systems. Research on this topic was motivated primarily by control problems for non-holonomic driftless mechanical systems-such as mobile robots. In this paper, we present recent results on exponential stabilization by time-varying homogeneous feedback, and investigate their applications to control systems with drift. Two general methods, originally developed for driftless systems, are presented. The first method relies on scaling asymptotically stabilizing feedback into exponential stabilizing feedback via a dilation. The second method applies a time-averaging technique to simplify the stability analysis of the closed-loop homogeneous system. These methods are demonstrated on two challenging stabilization problems : the planar rigid body with two forces and the spatial rigid body with one force and two torques.