  The experimental realization of quantum-degenerate Bose gases made of atoms
with sizeable magnetic dipole moments has created a new type of fluid, known as
a quantum ferrofluid, which combines the extraordinary properties of
superfluidity and ferrofluidity. A hallmark of superfluids is that they are
constrained to rotate through vortices with quantized circulation. In quantum
ferrofluids the long-range dipolar interactions add new ingredients by inducing
magnetostriction and instabilities, and also affect the structural properties
of vortices and vortex lattices. Here we give a review of the theory of
vortices in dipolar Bose-Einstein condensates, exploring the interplay of
magnetism with vorticity and contrasting this with the established behaviour in
non-dipolar condensates. We cover single vortex solutions, including structure,
energy and stability, vortex pairs, including interactions and dynamics, and
also vortex lattices. Our discussion is founded on the mean-field theory
provided by the dipolar Gross-Pitaevskii equation, ranging from analytic
treatments based on the Thomas-Fermi (hydrodynamic) and variational approaches
to full numerical simulations. Routes for generating vortices in dipolar
condensates are discussed, with particular attention paid to rotating
condensates, where surface instabilities drive the nucleation of vortices, and
lead to the emergence of rich and varied vortex lattice structures. We also
present an outlook, including potential extensions to degenerate Fermi gases,
quantum Hall physics, toroidal systems and the Berezinskii-Kosterlitz-Thouless
transition.
