  This article aims at discussing the cosmological constant problem at a
pedagogical but fully technical level. We review how the vacuum energy can be
regularized in flat and curved space-time and how it can be understood in terms
of Feynman bubble diagrams. In particular, we show that the properly
renormalized value of the zero-point energy density today (for a free theory)
is in fact far from being 122 orders of magnitude larger than the critical
energy density, as often quoted in the literature. We mainly consider the case
of scalar fields but also treat the cases of fermions and gauge bosons which
allows us to discuss the question of vacuum energy in super-symmetry. Then, we
discuss how the cosmological constant can be measured in cosmology and
constrained with experiments such as measurements of planet orbits in our solar
system or atomic spectra. We also review why the Lamb shift and the Casimir
effect seem to indicate that the quantum zero-point fluctuations are not an
artifact of the quantum field theory formalism. We investigate how experiments
on the universality of free fall can constrain the gravitational properties of
vacuum energy and we discuss the status of the weak equivalence principle in
quantum mechanics, in particular the Collela, Overhausser and Werner experiment
and the quantum Galileo experiment performed with a Salecker-Wigner-Peres
clock. Finally, we briefly conclude with a discussion on the solutions to the
cosmological constant problem that have been proposed so far.

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Phenomenology;High Energy Physics - Theory