Amplitude-modulation frequency-modulation (AM-FM) decompositions represent images using spatially-varying sinusoidal waves and their spatially-varying amplitudes. AM-FM decompositions use different scales and bandpass filters to extract the wide range of instantaneous frequencies and instantaneous amplitude components that may be present in an image. In the past few years, as the understanding of its theory advanced, AM-FM decompositions have been applied in a series of medical imaging problems ranging from ultrasound to retinal image analysis, yielding excellent results. This article summarizes the theory of AM-FM decompositions and related medical imaging applications. © 2012 Murray et al.