A Euclidean SU(2) monopole consists of a connection and Higgs field on an SU(2) bundle over π^{3}, satisfying certain partial differential equations. Monopoles may equivalently be described in terms of holomorphic vector bundles on twistor space, algebraic curves in twistor space, rational maps, or solutions to Nahm's equations (a set of ODEs for matrixvalued functions), all satisfying some further conditions. Research by Atiyah, Donaldson, Hitchin, Nahm and others has provided a beautiful and relatively complete picture of these different viewpoints and the links between them. Monopoles have also been studied on hyperbolic space π^{3}, although the corresponding picture in this case is less well understood. One difficulty is that the conditions which must be imposed in order for all the various correspondences to be valid have not yet been completely determined. A partial answer is given in Chapter 2, where it is proved that any hyperbolic monopole arising from a spectral curve satisfies a certain natural boundary condition. The proof uses the algebraic geometry of the spectral curve and is similar to Hurtubise's proof of the analogous result in the Euclidean case. A large part of this thesis concentrates on the "BraamAustin" description of hyperbolic monopoles. This is the hyperbolic version of Nahm's description of Euclidean monopoles; a monopole corresponds to a pair of discrete matrixvalued functions satisfying some difference equations. Euclidean monopoles appear as limits of hyperbolic monopoles as the curvature of π^{3} tends to zero. This "Euclidean limit" is described geometrically and is studied in terms of BraamAustin data. Explicit conditions are given for such a sequence to have a subsequence converging to a Euclidean monopole. The result depends on a conjecture (§ 4.5) about properties of BraamAustin monopole solutions.
