Fluid mixing generically refers to a cascading mechanism that transfers information to smaller and smaller spatial scales, in a way that is time reversible and conservative for finite times but results in an irreversible loss of information at in infinite time. In simple terms, mixing is what is observed when stirring two liquids together, resulting in the creation of a homogenized mixture, but it can also be thought as a more complicated stabilizing mechanism for certain stationary structures, generating damping effects. In this mini-course I will review several aspects linking Fourier analysis, spectral theory and fluid mixing, in the context of advection-diffusion equations driven by incompressible velocity fields, and also related to hydrodynamic stability problems in the two-dimensional Navier-Stokes equations.