  We describe basic concepts of noncommutative geometry and a general
construction extending the familiar duality between ordinary spaces and
commutative algebras to a duality between Quotient spaces and Noncommutative
algebras. Basic tools of the theory, K-theory, Cyclic cohomology, Morita
equivalence, Operator theoretic index theorems, Hopf algebra symmetry are
reviewed. They cover the global aspects of noncommutative spaces, such as the
transformation $\theta \to 1/\theta$ for the NC torus $\Tb_{\theta}^2$, unseen
in perturbative expansions in $\theta$ such as star or Moyal products. We
discuss the foundational problem of "what is a manifold in NCG" and explain the
role of Poincare duality in K-homology which is the basic reason for the
spectral point of view. When specializing to 4-geometries this leads to the
universal "Instanton algebra". We describe our work with G. Landi which gives
NC-spheres $S_{\theta}^4$ from representations of the Instanton algebra. We
show that any compact Riemannian spin manifold whose isometry group has rank $r
\geq 2$ admits isospectral deformations to noncommutative geometries. We give a
survey of our work with H. Moscovici on the transverse geometry of foliations
which yields a diffeomorphism invariant geometry on the bundle of metrics on a
manifold and a natural extension of cyclic cohomology to Hopf algebras. Then,
our work with D. Kreimer on renormalization and the Riemann-Hilbert problem.
Finally we describe the spectral realization of zeros of zeta and L-functions
from the noncommutative space of Adele classes on a global field and its
relation with the Arthur-Selberg trace formula in the Langlands program. We end
with a tentalizing connection between the renormalization group and the missing
Galois theory at Archimedian places.
