The complexity of multiple vortex fields requires an adequate mathematical foundation. We present a modal theory based on Fourier mathematics using cylindrical coordinates. The analysis is analogous to well-known 2D Fourier optics formalism using Cartesian coordinates. The theory is comprehensive in the sense that it cannot only be used to describe and analyze arbitrary vortex fields, but it may also be used for their synthesis. Here, after discussing the basic formalism, we consider various examples of vortex fields, as well as special topics such as the algorithmic design of vortex fields by using computer-generated diffractive optics, self-imaging, and nano-vortices.