Cuntz semigroups – the probable future of the classification program for $C^*$-algebras

$K$-theory is the standard invariant for $C^*$-algebras, but many properties of the algebra, such as the ideal lattice or topological dimension, aren't reflected in $K$-theory. Initially Cuntz semigroups were introduced (by Cuntz himself) to construct quasi-traces on stably finite simple $C^*$-algebras, but since then they have become a popular invariant with a long list of applications. The definition of this invariant is similar to the definition of $K$-theory, but instead of projections we consider all positive elements! I will give several different definitions of Cuntz semigroups and talk about their most striking properties and applications.