Triangulated surfaces are compact hyperbolic Riemann surfaces that admit a conformal triangulation by equilateral triangles. Brooks and Makover started the study of random triangulated surfaces in the large genus limit, and proved results about their systole, diameter and Cheeger constant. Subsequently Mirzakhani proved analogous results about random hyperbolic surfaces. These results, along with many others, suggest that the large genus geometry of random triangulated surfaces mirrors the large genus geometry of random hyperbolic surfaces. In this talk, I will describe an approach to show that triangulated surfaces are well distributed in moduli space in the large genus limit. Speaker(s): Sahana Vasudevan (MIT)