Abstract: Geostrophic motions and internal gravity waves form two of the primary building blocks of geophysical fluid dynamics and methods for disentangling these types of motions are often referred to as wave-vortex decompositions. In this talk we clarify the landscape of decompositions and discuss why such decompositions are so important to our understanding of geophysical flows. We discuss the theory, including the role of available potential vorticity, and show a number of examples of how such decompositions can be used.