A survey on topological properties of $P(K)$ spaces

Given a compact space $K$, we denote by $P(K)$ the space of all Radon probability measures on $K$, equipped with the $wea{k}^{\ast }$ topology inherited from $C(K{)}^{\ast }$. For nonmetrizable compacta $K$ even basic properties of $P(K)$ spaces depend of additional axioms of set theory. We discuss here older and quite recent results on the subject.