A vortex is generally regarded as a flow in which fluid particles swirl around some central line, which is termed the vortex axis. Most dictionaries provide definitions of vortex that describe it as a whirlpool or swirling flow, emphasizing the rotational characteristic of vortices. Other definitions provided in dictionaries mention the low pressure typically found at the vortex center and the presence of angular velocity of the fluid elements (or vorticityVorticity). The problem with defining a vortex based on swirling motion is that what appears as swirl to one observer will not appear as swirl to another observer. Clearly the identification of a flow as a vortex must be independent of the motion of the person observing the flow! The problem with defining a vortex based on low pressure or angular rotation of fluid elements is that, while these are features of vortices, they are also features of other flows that are clearly not vortices. None of these definitions are sufficient to define a vortex; nevertheless, they serve to illustrate some of the popular conceptions of vortices. The question “What is a vortex?” is addressed in this chapter from three points of view—by example (Sect. 1.1), with reference to the properties of a vortex (Sect. 1.2), and mathematically (Sect. 1.3). As we shall see, the question has unexpected depth.