Quantum Spaces : Poincaré Seminar 2007 This book starts with an introduction by V. Pasquier on the usefulness of non commutative geometry especially in the condensed matter context of the Hall effect. This theme is further developed by A. Polychronakos who together with L. Susskind introduced the concept of non commutative fluids. Jean Michel Maillet compares the experimental results on one dimensional magnetic chains to the theoretical predictions based on the algebraic Bethe Ansatz and related to quantum group symmetries that he actively developed with many collaborators. On the high energy side of non commutative geometry the book includes two complementary reviews. A. Connes describes the striking progress recently made in gathering all the interactions and fields of the standard model into a non commutative geometry on a simple internal space while V. Rivasseau describes in a large review the very recent technique of renormalization of quantum field theories on non commutative space time which lead to the surprising discovery of their improved short distance behavior. This book confirms noncommutative geometry as an increasingly useful tool for the description of intricate condensed matter phenomena. It describes the striking progress recently made in gathering all the interactions and fields of the standard model into a non commutative geometry on a simple internal space. Coverage also details the very recent technique of renormalization of quantum field theories on non commutative space time.