Let \(G\) be a countably infinite group and consider the class of all measure-preserving actions of \(G\) on probability spaces. The best method for understanding this class, according to ideas from continuous model theory and particularly Robinson's philosophy, is to study the probability-measure-preserving actions which are "existentially closed." Roughly, these are the actions which possess "arbitrarily good solutions" to all "equations" which "should have" solutions. In this talk, I will define in more detail what this means and provide some examples. Then I will discuss how our ability to understand existentially closed actions hinges upon topological phenomena, particularly first cohomology groups. This work leaves open a vexing question: are the existentially closed actions axiomatizable for every countable group \(G\)? The results presented are joint work with Isaac Goldbring and Robin Tucker-Drob.