While not all regions of a flow that contain vorticity are vortices, all vortices contain vorticity. Vorticity transport and the laws that regulate it are therefore fundamental to the study of vortices. This chapter develops the mathematical theory of vorticity transport, with a focus on incompressible flows. Kinematical constructs used to describe vortex flows, such as vortex lines and vortex tubes, are first defined. The Helmholtz decomposition is applied to the velocity vector, and the vector potential is defined. A Green's function approach is used to solve the Poisson equation for vector potential, leading to derivation of the Biot-Savart equation for the velocity field induced by vorticity. The vorticity transport equation is derived, and Cauchy's general solution for inviscid fluids is presented. The Helmholtz vortex laws and the flow invariants for both three- and two-dimensional flows are developed, which together provide a foundation for the theory of vortex flows.