  We give a detailed introduction to the theory of Cuntz semigroups for
C*-algebras. Beginning with the most basic definitions and technical lemmas, we
present several results of historical importance, such as Cuntz's theorem on
the existence of quasitraces, R{\o}rdam's proof that $\mathcal{Z}$-stability
implies strict comparison, and Toms' example of a non $\mathcal{Z}$-stable
simple, nuclear C*-algebra. We also give the reader an extensive overview of
the state of the art and the modern approach to the theory, including the
recent results for C*-algebras of stable rank one (for example, the
Blackadar-Handelman conjecture and the realization of ranks), as well as the
abstract study of the Cuntz category $\mathbf{Cu}$.
