In physical models of fluids, a singularity occurs when a quantity of interest (velocity, pressure, vorticity…) becomes infinite at some finite time, starting from a "nice" initial configuration. In this talk, we will first introduce some classical models of compressible and incompressible fluids. We will then describe several mechanisms by which fluids can form singularities in finite time, allowing us to discuss the singularity formation problem for the incompressible three-dimensional Euler equations and the Navier--Stokes equations. We will finally touch upon computer-assisted methods and their applications to singularity formation in fluids.