This paper presents analysis and design results for control systems subject to oscillatory inputs, i.e., inputs of large amplitude and high frequency. The key analysis results are a series expansion characterizing the averaged system and various Lie-algebraic conditions that guarantee the series can be summed. Various example systems provide insight into the results. With regards to design, we recover and extend a variety of point stabilization and trajectory tracking results using oscillatory controls. We present novel developments on stabilization of systems with positive trace and on tracking for second order underactuated systems.