Groupoids form a convenient framework to study various kinds of (topological) dynamical systems. They also play an increasingly important role in the abstract theory of C*-algebras and their classification. This talk will focus on homological and K-theoretical invariants of groupoids. In particular, I will give an overview on some recent progress on a conjecture posed by Hiroki Matui that relates the K-theory of a groupoid C*-algebra to the homology of the underlying groupoid itself.