It is a classical fact that countable groups of homeomorphisms of the interval and the circle are characterized by purely algebraic properties, namely linear and circular orderability. It remains a difficult and deep problem to understand countable groups that can act by homeomorphisms on compact manifolds in dimension two or more. I will discuss some new ways of investigating groups acting on manifolds through ideas from mathematical logic and descriptive set theory. This talk will include work that is joint with Sang-hyun Kim, J. De la Nuez Gonzalez, and Yash Lodha.