  We first review the historical developments, both in physics and in
mathematics, that preceded (and in some sense provided the background of)
deformation quantization. Then we describe the birth of the latter theory and
its evolution in the past twenty years, insisting on the main conceptual
developments and keeping here as much as possible on the physical side. For the
physical part the accent is put on its relations to, and relevance for,
"conventional" physics. For the mathematical part we concentrate on the
questions of existence and equivalence, including most recent developments for
general Poisson manifolds; we touch also noncommutative geometry and index
theorems, and relations with group theory, including quantum groups. An
extensive (though very incomplete) bibliography is appended and includes
background mathematical literature.
