We discuss the physics of vortices in the circumstellar disks associated with young stellar objects. We elucidate the basic physical properties of these localized storm systems. In particular, we consider point vortices, linear vortices, the effects of self-gravity, magnetic fields, and nonlinear aspects of the problem. We find that these vortices can exist in many different forms in the disks of young stellar objects and may play a role in the formation of binary companions and/or giant planets. Vortices may enhance giant planet formation via gravitational instability by allowing dust grains (heavy elements) to settle to the center on a short timescale; the gravitational instability itself is also enhanced because the vortices also create a larger local surface density in the disk. In addition, vortices can enhance energy dissipation in disks and thereby affect disk accretion. Finally, we consider the possibility that vortices of this type exist in molecular clouds and in the disk of the galaxy itself. On all of these size scales, vortices can produce long-lived structures which may correspond to observed structures in these systems