An introduction to six-functor formalisms

These are notes for a mini-course given at the summer school and conference "The Six-Functor Formalism and Motivic Homotopy Theory" in Milan 9/2021. They provide an introduction to the formalism of Grothendieck's six operations in algebraic geometry and end with an excursion to rigid-analytic motives. The notes do not correspond precisely to the lectures delivered but provide a more self-contained account for the benefit of the audience and others. No originality is claimed.