  This thesis was motivated by a desire to understand the natural geometry of
hyperbolic monopole moduli spaces. We take two approaches. Firstly we develop
the twistor theory of singular hyperbolic monopoles and use it to study the
geometry of their charge 1 moduli spaces. After this we introduce a new way to
study the moduli spaces of both Euclidean and hyperbolic monopoles by applying
Kodaira's deformation theory to the spectral curve. We obtain new results in
both the Euclidean and hyperbolic cases. In particular we prove new cohomology
vanishing theorems and find that the hyperbolic monopole moduli space appears
to carry a new type of geometry whose complexification is similar to the
complexification of hyperk\"ahler geometry but with different reality
conditions.
