The divisor class group of a hyperelliptic curve over a field is a finite abelian group at the center of many open questions in algebraic geometry and number theory. Sutherland surveys some of these, including the computation of the associated L-functions and zeta functions used in his investigation of Sato-Tate distributions. Many of these problems lend themselves to empirical investigation, and as emphasized by Sutherland, fast arithmetic in the divisor class group is crucial for their efficiency. Indeed, implementations of these fundamental operations are at the core of the algebraic geometry packages of widely-used computer algebra systems such as Magma and Sage. This thesis provides contributions to improve the efficiency of divisor class arithmetic on hyperelliptic curves, with special attention to the often ignored split model cases and to genus 2 and 3. There are two main contributions: the introduction of “Balanced NUCOMP” for improved arithmetic on curves given by split models of arbitrary genus, and improved explicit formulas for genus 2 (ramified and split models) and genus 3 (split models) based on Balanced NUCOMP for split models and NUCOMP for ramified models. Empirical analysis, using a complete Magma implementation and testing suite, is conducted with all contributed algorithms to provide proof of correctness and comparisons to previous best. Our results show that Balanced NUCOMP does offer improvements to split model arithmetic, further narrowing the performance gap between split and ramified models. Our explicit formulas require fewer field operations, most notably significantly fewer additions, than previous best for computing arithmetic on genus 2 (ramified and split models), and genus 3 (split models). Our empirical results demonstrate that our formulas yield the fastest general-purpose arithmetic for these cases.