In this talk, we discuss various algorithmic techniques used for counting and sampling subgraphs in a large input graph. The focus of the talk is on the mathematical foundations. We start with the beautiful technique of Color-Coding (Alon, Yuster, Zwick 1995), and we discuss various generalizations based on group algebras (Koutis 2008) and on the exterior algebra (Brand, D, Husfeldt 2018). These techniques are most useful for sampling, which is equivalent to approximate counting. On the other hand, the fastest known algorithm to exactly count subgraphs that are isomorphic to a graph H  (Curticapean, D, Marx 2017) is based on the foundations of Lovász' theory of graph limits.