It is well-known that the moduli space of genus g curves with n marked points, denoted M_g,n, can be defined over the rational numbers Q. In this talk, we will discuss arithmetic local systems over M_g,n (i.e. representations of the arithmetic etale fundamental group of M_g,n) and their relation to Galois representations. While there are many arithmetic local systems of more obvious origins, there is also an exotic family of local systems that can be constructed using the ideas of physics (conformal field theory).