The electronic properties of graphene, a two-dimensional crystal of carbon atoms, are exceptionally novel. For instance, the low-energy quasiparticles in graphene behave as massless chiral Dirac fermions which has led to the experimental observation of many interesting effects similar to those predicted in the relativistic regime. Graphene also has immense potential to be a key ingredient of new devices, such as single molecule gas sensors, ballistic transistors and spintronic devices. Bilayer graphene, which consists of two stacked monolayers and where the quasiparticles are massive chiral fermions, has a quadratic low-energy band structure which generates very different scattering properties from those of the monolayer. It also presents the unique property that a tunable band gap can be opened and controlled easily by a top gate. These properties have made bilayer graphene a subject of intense interest. In this review, we provide an in-depth description of the physics of monolayer and bilayer graphene from a theorist's perspective. We discuss the physical properties of graphene in an external magnetic field, reflecting the chiral nature of the quasiparticles near the Dirac point with a Landau level at zero energy. We address the unique integer quantum Hall effects, the role of electron correlations, and the recent observation of the fractional quantum Hall effect in the monolayer graphene. The quantum Hall effect in bilayer graphene is fundamentally different from that of a monolayer, reflecting the unique band structure of this system. The theory of transport in the absence of an external magnetic field is discussed in detail, along with the role of disorder studied in various theoretical models. Recent experminental observations of a metal-insulator transition in hydrogenated graphene is discussed in terms of a self-consistent theory and compared with related numerical simulations. We highlight the differences and similarities between monolayer and bilayer graphene, and focus on thermodynamic properties such as the compressibility, the plasmon spectra, the weak localization correction, quantum Hall effect and optical properties. Confinement of electrons in graphene is non-trivial due to Klein tunnelling. We review various theoretical and experimental studies of quantum confined structures made from graphene. The band structure of graphene nanoribbons and the role of the sublattice symmetry, edge geometry and the size of the nanoribbon on the electronic and magnetic properties are very active areas of research, and a detailed review of these topics is presented. Also, the effects of substrate interactions, adsorbed atoms, lattice defects and doping on the band structure of finite-sized graphene systems are discussed. We also include a brief description of graphane-gapped material obtained from graphene by attaching hydrogen atoms to each carbon atom in the lattice.