This 4-session talk is meant as an introduction to the theory of Cuntz semigroups and their abstract counterparts, known as Cu-semigroups. During the first session I will review some of the most important aspects of the invariant, such as its ideal lattice, its functionals, and its target category $\mathrm{Cu}$.

In the other three sessions I will present some recent results (all based on joint work with H. Thiel) on three different aspects of the Cuntz semigroup: Softness, nowhere scatteredness and the Global Glimm Property, and covering dimension.