In this review, we will focus on the main problems of point vortices theory and deal with issues related to their interaction with each other and with vortex spots, as well as describe the main problems in the theory of vortex chains and lattices in an ideal vortex fluid. A distinctive feature of the presentation is the systematic development of mathematical methods associated with the penetration into the vortex dynamics of the main ideas of the theory of Poisson structures and Lie algebras, topology, bifurcation and qualitative analysis. Particular attention in the review is paid to the problems of integrability, non-integrability and qualitative analysis of dynamic systems of vortex dynamics.