A two-dimensional idealization is often best suited for examining certain phenomena exhibited by vortex flows. This chapter examines classical vortex structures, such as the Rankine vortex and the Gaussian vortex, and introduces the limiting case of a point vortex. Point vortex systems are examined, including vortex pair and vortex polygons, as well as invariants and stability of point vortex systems. Vortex sheets are introduced as the result of a limiting process of a discrete vortex array. The Birkhoff-Rott equation for vortex sheet motion is derived, and it is used to derive the Kelvin-Helmholtz instability and the Kaden spiral. We then turn to dynamics of vortex patches, including the Kirchhoff solution for an elliptical vortex patch and solutions for motion of an elliptical vortex patch in straining and shearing flows. The chapter ends with discussion of the phenomenon of merger of two co-rotating vortices.