This thesis describes a number of algorithms and properties relating to Gromov’s word-hyperbolic groups. A fuller outline of the thesis is given, and a number of basic concepts relating to metric spaces, hyperbolicity and automaticity are first briefly detailed in Chapter 1. Chapter 2 then details a solution to the conjugacy problem for lists of elements in a word-hyperbolic group which can be run in linear time; this is an improvement on a quadratic time algorithm for lists which contain an infinite order element. Chapter 3 provides a number of further results and algorithms which build upon this result to efficiently solve problems relating to quasiconvex subgroups of word-hyperbolic groups – specifically, the problem of testing if an element conjugates into a quasiconvex subgroup, and testing equality of double cosets. In Chapter 4, a number of properties of certain coset Cayley graphs are studied, in particular showing that graph morphisms which preserve edge labels and directions and map a quasiconvex subset to a single point also preserve a variety of other properties, for instance hyperbolicity. Finally, Chapter 5 gives a proof that all word-hyperbolic groups are 14-hyperbolic with respect to some generating set.