Arithmetical structures on finite connected graphs are generalization of the Laplacian of a graph. Dino Lorenzini originally defined them in order to answer some questions in algebraic geometry, but more recently, they have been studied on their own, particularly with a combinatorics lens. In this talk, we will discuss how to count the number of arithmetical structures on different types of graphs and discuss why it is a hard but interesting question for other families. If time permits, we will talk about their corresponding critical groups.