  In this paper, we develop a theoretical framework for time-harmonic wave
scattering on hyperbolic spaces. Using the limiting absorption principle (LAP),
we derive the explicit forms of the ingoing and outgoing Green functions of the
Helmholtz operator of hyperbolic spaces, and verify that they are the
fundamental solutions. Then we establish accurate characterisations of the
asymptotic behaviours of the Green functions and use them to establish the
ingoing and outgoing radiation conditions, which are analogues to the
Sommerfeld radiation conditions in the Euclidean setting. Moreover, we prove a
Rellich's type theorem which guarantees that the scattered field as well as its
far-field pattern are uniquely defined. Within the framework, we consider the
scattering from a source and a potential respectively. To our best knowledge,
the theoretical framework is new to the literature and it paves the way for
many subsequent developments for wave scattering on hyperbolic spaces.
