Topological dynamics is the study of continuous group actions. In this context, the best understood groups are the Abelian ones, especially $\mathbb{Z}$, while the general theory remains far less clear. However, recent years saw the emergence of powerful methods that often work without any assumptions on the group at all. Instead of group theory, these methods are based on techniques from combinatorics and descriptive set theory. I will illustrate this development by addressing the following very basic but surprisingly challenging problem: If $\Delta$ is a subgroup of $\Gamma$, does $\Gamma$ have a free continuous action on a compact space without any nontrivial closed $\Delta$-invariant subsets? Based on joint work with Joshua Frisch.