Medical image analysis methods require the use of effective representations for differentiating between lesions, diseased regions, and normal structure. Amplitude Modulation-Frequency Modulation (AM-FM) models provide effective representations through physically meaningful descriptors of complex non-stationary structures that can differentiate between the different lesions and normal structure. Based on AM-FM models, medical images are decomposed into AM-FM components where the instantaneous frequency provides a descriptor of local texture, the instantaneous amplitude captures slowly-varying brightness variations, while the instantaneous phase provides for a powerful descriptor of location, generalizing the traditionally important role of phase in the Fourier Analysis of images. Over the years, AM-FM representations have been used in a wide variety of medical image analysis applications based on a vastly reduced number of features that can be easily learned by simple classifiers. The paper provides an overview of AM-FM models and methods, followed by applications in medical image analysis. We also provide a summary of emerging trends and future directions.