Interactions of vortex structures play an important role in the understanding of complex evolutions of fluid flows. Incompressible and inviscid flows with pointwise singular vorticity distributions in two-dimensional space, called point vortices, have been used as a theoretical model to describe such vortex interactions. In particular, the equilibrium states of these point vortices are called vortex crystals. Vortex crystals are mathematically formulated based on some elliptic partial differential equations for stream function. From the application point of view, vortex crystals are not only considered as an intrinsic theoretical model of flows in fluid mechanics. In addition, they also appear in many physical systems such as quantum mechanics and flows of superfluid films. In this talk, after reviewing some mathematical aspects of the theory of vortex dynamics, we introduce recent studies about vortex crystals on closed surfaces such as a sphere, a curved torus, and a flat torus (i.e., a doubly periodic plane), on which the background smooth vorticity distributions such as a constant vorticity distribution and a Liouville-type vorticity distribution, where an exponential function of the stream function represents the vorticity.