Around 2000, in their "Contact topology and hydrodynamics" saga, Etnyre and Ghrist established a correspondence between Reeb and Beltrami fields (the non-vanishing solutions to the steady Euler equations that are eigenfields of the curl operator), which allowed them to import tools from contact topology to the hydrodynamics governed by the Euler equations. For example, they studied the existence of closed flowlines of said equations on certain manifolds, as well as the types of knots these flowlines can realize. After that, around 2012-2015, analysts Enciso and Peralta-Salas proved stronger results on flowlines and vortex tubes. More recently, the field has been seeing a revival of the use of contact topological methods, with results winking at the "universality of Euler equations" program of Tao. I will try to paint a broad picture of all of the above, assuming no familiarity with the analytical side (i.e. I will introduce the Euler equations), and minimal familiarity with the contact topological side.