A formal group law is a symmetric function of the form $f(f^{-1}(x_1) + f^{-1}(x_2) + \cdots )$ for a power series $f(x)$. A chromatic symmetric function of a hypergraph is a generalization, due to Stanley, of the chromatic polynomial of a graph counting proper colorings. In fact, there is a connection between these seemingly unrelated objects. We will show that in many cases, if $f(x)$ is a generating function for a class of combinatorial objects then the associated formal group law is a sum of chromatic symmetric functions of hypergraphs. Examples include graphs, permutations, lattice paths, and various types of trees.