In graphs and surfaces, as well as in other categories of geometric structures, a natural and fruitful model for a random object is that of a random topological cover of a fixed base object. A random degree-n covering space of a given graph or surface G is equivalent to a random homomorphism from the fundamental group of G to the symmetric group S_n. 

I will explain these models, what we know about properties of the random cover as a whole, and also what we know about the behavior in the cover of a given fixed loop in the base object. As I will show, there is a lot of surprising mathematical structure hiding in these questions.

I will also touch upon variants of these questions. For example, what happens if we take random homomorphisms into the unitary group instead (this is very much related to Yang-Mills theory), or what structure we get if we take random homomorphisms into other groups like GL_N(q). We will see that a variety of mathematics appears around these questions: from topology and combinatorics, to representation theory and algebra.