This project is centred around the study of surfaces, in particular hyperbolic surfaces, and their moduli spaces. These objects are ubiquitous throughout mathematics and are either an integral part of or use techniques from a host of different types of geometry, topology, combinatorics, analysis and number theory. In simple terms, the project revolves around the study of curves and arcs on surfaces, and to use the developed techniques to understand moduli type spaces and their related groups.There are three main themes to the project. The first theme is the study of orthogeodesics and related identities. One of the specific goals is to prove length rigidity results for orthogeodesics. The second theme is about the relationship between diameters and systoles of surfaces, and is related to finding upper bounds on systole length. The third theme is more combinatorial, and is about how to relate the geometry and combinatorics of curves and arcs to moduli type spaces, and in particular to the understanding of the mapping class group.