In this thesis we discuss two related and important objects in the study of geometric group theory and Teichmüller theory, namely the 'curve complex' and 'mapping class group'. The original material is entirely contained in Chapter 2 and Chapter 3. the two chapters are self-contained and can be read independently, and independent of the introduction in Chapter 1. Both chapters form part of my long-term interest in the computational and large-scale geometry of the curve complex and in the structure of the mapping class group. In Chapter 2, we discuss some of the computability aspects of the path-metric on the 1-skeleton of the curve complex, called the 'curve graph'. Specifically, we develop an algorithm for computing distances in the curve graph by constructing (all) tight geodesics between any two of its vertices. Our work produces bounds on intersection numbers associated to tight geodesics without the need for taking geometric limits. Thus, we complement the work of Masur-Minsky and of Bowditch in this area. We then discuss some of the implications for the action of the 'mapping class group' on the new graph. We discover a computability version of the acylindricity theorem of Bowditch and we recover all of the weak proper discontinuity of Bestvina-Fujiwara. Our methods are entirely combinatorial. In Chapter 3, we study combinatorial rigidity questions regarding the curve complex and the mapping class group. We see that every embedding of one curve complex to itself or another, whose dimensions do not increase from domain to codomain, is induced by a surface homeomorphism. While previous approaches due to Ivanov (for automorphisms) and Irmak (for superinjective maps) for the same surface require a deep result proven by Harer and by Bowditch-Epstein, among others, regarding the existence of a triangulation of 'Teichmüller space' and the approach due to Luo makes essential use of a modular structure, we shall require little more than the connectivity of links in the curve complex.