Liouville quantum gravity is a certain canonical random geometry in two dimensions. In this talk we will discuss the spectral geometry of the associated Laplace-Beltrami operator. After showing that the eigenvalues a.s. obey a Weyl law, we will discuss a surprising connection between asymptotics of the heat trace and the KPZ (Knizhnik-Polyakov-Zamolodchikov) scaling theory. I also plan to discuss a number of conjectures.  
 The talk will be expository in nature; there will be no proofs and no background will be assumed about LQG.