  The cosmological constant problem is one of the long-standing issues of
modern physics. While we can measure the value of the cosmological constant
with great accuracy, we are not able to calculate it in a coherent theoretical
framework. On the contrary the theoretical predictions in Quantum Field Theory
are radically different from observations. This disagreement is a hint of the
difficult conciliation of Quantum Mechanics and General Relativity in a theory
of Quantum Gravity. Current approaches to the cosmological constant problem, in
particular, do not account for the quantum nature of the gravitational
interaction and rely on perturbative calculations. In this thesis we address
the issue in the simplified framework of two-dimensional dilaton-Maxwell
gravity, coupled to scalar matter fields. In this setting we are able to
quantize our model non-perturbatively in Dirac's approach to constrained
systems. We determine that the realization of the classical symmetries at the
quantum level provides a mechanism that fixes the value of the cosmological
constant once a specific quantum state of the Universe is selected. Furthermore
Quantum Gravity introduces opposite contributions to the cosmological constant,
admitting a range of values compatible with current observations.
