The study of vortex filaments in fluids dates back to Kelvin and Helmholtz in the 19th century. In these lectures, I will first present the problem from an historical perspective and some mathematical questions (most of which are still considered open) that have been associated with it over time, in particular regarding the description of their dynamics in the case of thin vortex cores. I will then describe joint works with Thierry Gallay (for the Euler equation) and Robert L. Jerrard (for the Gross-Pitaevskii equation) that have at partially answered some of these questions, and finally will try to explain (our understanding of) the difficulties that remain to be solved.