  The present notes provide an extended version of a small lecture course given
at the Humboldt Universit\"at zu Berlin in the Winter Term 2022/23 (of 36
hours). The material starting in Section 5.4 was added afterwards. The aim of
these notes to give an introductory overview on the analytical approaches for
gradient-flow equations in Hilbert spaces, Banach spaces, and metric spaces and
to show that on the first entry level these theories have a lot in common. The
theories and their specific setups are illustrated by suitable examples and
counterexamples.
