  The first part of my thesis lays the foundations to generalized Lorentz
geometry. The basic algebraic structure of finite-dimensional modules over the
ring of generalized numbers is investigated. The motivation for this part of my
thesis evolved from the main topic, the wave equation on singular space-times.
The second and main part of my thesis is devoted to establishing a local
existence and uniqueness theorem for the wave equation on singular space-times.
The singular Lorentz metric subject to our discussion is modeled within the
special algebra on manifolds in the sense of Colombeau. Inspired by an approach
to generalized hyperbolicity of conical-space times due to Vickers and Wilson,
we succeed in establishing certain energy estimates, which by a further
elaborated equivalence of energy integrals and Sobolev norms allow us to prove
existence and uniqueness of local generalized solutions of the wave equation
with respect to a wide class of generalized metrics. The third part of my
thesis treats three different point value resp. uniqueness questions in
algebras of generalized functions
