Dirac Matter This fifteenth volume of the Poincare Seminar Series Dirac Matter describes the surprising resurgence as a low energy effective theory of conducting electrons in many condensed matter systems including graphene and topological insulators of the famous equation originally invented by P.A.M. Dirac for relativistic quantum mechanics. In five highly pedagogical articles as befits their origin in lectures to a broad scientific audience this book explains why Dirac matters. Highlights include the detailed Graphene and Relativistic Quantum Physics written by the experimental pioneer Philip Kim and devoted to graphene a form of carbon crystallized in a two dimensional hexagonal lattice from its discovery in 2004 2005 by the future Nobel prize winners Kostya Novoselov and Andre Geim to the so called relativistic quantum Hall effect; the review entitled Dirac Fermions in Condensed Matter and Beyond written by two prominent theoreticians Mark Goerbig and Gilles Montambaux who consider many other materials than graphene collectively known as Dirac matter and offer a thorough description of the merging transition of Dirac cones that occurs in the energy spectrum in various experiments involving stretching of the microscopic hexagonal lattice; the third contribution entitled Quantum Transport in Graphene: Impurity Scattering as a Probe of the Dirac Spectrum given by Hélène Bouchiat a leading experimentalist in mesoscopic physics with Sophie Guéron and Chuan Li shows how measuring electrical transport in particular magneto transport in real graphene devices contaminated by impurities and hence exhibiting a diffusive regime allows one to deeply probe the Dirac nature of electrons. The last two contributions focus on topological insulators; in the authoritative Experimental Signatures of Topological Insulators Laurent Lévy reviews recent experimental progress in the physics of mercury telluride samples under strain which demonstrates that the surface of a three dimensional topological insulator hosts a two dimensional massless Dirac metal; the illuminating final contribution by David Carpentier entitled Topology of Bands in Solids: From Insulators to Dirac Matter provides a geometric description of Bloch wave functions in terms of Berry phases and parallel transport and of their topological classification in terms of invariants such as Chern numbers and ends with a perspective on three dimensional semi metals as described b ... Philip Kim: Graphene and Relativistic Quantum Physics Mark Goerbig and Gilles Montambaux : Dirac Fermions in Condensed Matter and Beyond Chuan Li Sophie Guéron Hélène Bouchiat : Quantum Transport in Graphene : Impurity Scattering as a Probe of the Dirac Spectrum. Laurent Lévy : Experimental Signatures of Topological Insulators David Carpentier :Topology of Bands in Solids: From Insulators to Dirac Matter