  This Ph.D. thesis pursues two goals: The study of the geometrical structure
of two-dimensional quantum gravity and in particular its fractal nature. To
address these questions we review the continuum formalism of quantum gravity
with special focus on the scaling properties of the theory. We discuss several
concepts of fractal dimensions which characterize the extrinsic and intrinsic
geometry of quantum gravity. This work is partly based on work done in
collaboration with Jan Ambj{\o}rn, Dimitrij Boulatov, Jakob L. Nielsen and
Yoshiyuki Watabiki (1997).
  The other goal is the discussion of the discretization of quantum gravity and
to address the so called quantum failure of Regge calculus. We review dynamical
triangulations and show that it agrees with the continuum theory in two
dimensions. Then we discuss Regge calculus and prove that a continuum limit
cannot be taken in a sensible way and that it does not reproduce continuum
results. This work is partly based on work done in collaboration with Jan
Ambj{\o}rn, Jakob L. Nielsen and George Savvidy (1997).
