  The existence of dark matter was suggested, using simple gravitational
arguments, seventy years ago. Although we are now convinced that most of the
mass in the Universe is indeed some non-luminous matter, we still do not know
its composition. The problem of the dark matter in the Universe is reviewed
here. Particle candidates for dark matter are discussed with particular
emphasis on Weakly Interacting Massive Particles (WIMPs). Experiments searching
for these relic particles, carried out by many groups around the world, are
also reviewed, paying special attention to their direct detection by observing
the elastic scattering on target nuclei through nuclear recoils. Finally, we
concentrate on the theoretical models predicting WIMPs, and in particular on
supersymmetric extensions of the standard model, where the leading candidate
for WIMP, the neutralino, is present. There, we compute the cross section for
the direct detection of neutralinos, and compare it with the sensitivity of
detectors. We mainly discuss supergravity, superstring and M-theory scenarios.
