This thesis studies the possibility of the quantum avoidance of gravitational singularities in
anisotropic cosmological models.
For that purpose, we review the fundamentals of spatially homogeneous cosmological
models and quantum cosmology based on the Wheeler-DeWitt equation in minisuperspace.
Furthermore, we introduce a generalized dynamical system which is designed to emulate some
of the main features of the cosmological models. After studying its geometric properties, we
start to investigate how one can approach the canonical quantization of such a system. The
main focus of our analysis is on the factor ordering problem in the Wheeler-DeWitt equation.
The considerations motivate us to formulate criteria for singularity avoidance, that respect
the conformal geometry of the con�guration space of the spatially homogeneous models.
We then go on by studying some speci�c models with and without matter. In particular
we examine classical and quantum properties of the Bianchi type I, II and IX and the
Kantowski-Sachs universe. The criteria we developed previously are applied to see under
which circumstances singularities can be avoided. If the potential terms are negligible when
compared against the velocity terms in the gravitational action, the approach towards the
singularity is called asymptotically velocity term dominated. We �nd that such singularities
can be resolved, if the dimension of the minisuperspace is suffciently large. The underlying
mechanism is a spreading of wave packets in minisuperspace.
We also consider the non-diagonal Bianchi IX model with tilted dust. This model is
relevant in the context of the BKL scenario. We pay particular attention to the asymptotic
regime close to the singularity and the temporal behavior of curvature invariants in this
regime