Graphene, recently (2004) discovered new allotrope of carbon, is the first example of truly two-dimensional crystal consisting from just one layer of atoms. Electrons and holes in this material have properties similar toultrarelativistic particles (two-dimensional analog of massless Dirac fermions). This leads to some unusual and even counterintuitive phenomena, such as finite conductivity in the limit of zero charge carrier concentration (quantum transport by evanescent waves) or transmission of electrons through high and broad poential barriers with a high probability (Klein tunneling). This allows to study subtle effects of relativistic quantum mechanics and quantum field theory in condensed-matter experiments, without accelerators and colliders. Some of these effects were considered as practically unreachable. Apart from the Klein tunneling, this is, for example, a vacuum reconstruction near supercritical charges predicted many years ago for collisions of ultra-heavy ions. Another interesting class of quantum-relativistic phenomena is related with corrugations of graphene, which are unavoidable for any two-dimensional systems at finite temperature. As a result, one has not just massless Dirac fermions but massless Dirac fermions in curved space. Gauge fields, of the central concepts of modern physics, are quite real in graphene and one can manipulate them just applying mechanical stress.