The electronic properties of graphene, a two-dimensional crystal of carbon atoms, are unique and novel. Graphene exhibits massless chiral Dirac fermions, leading to interesting effects such as the quantum Hall effect and fractional quantum Hall effect. Bilayer graphene, composed of two stacked monolayers, has a quadratic low-energy band structure with different scattering properties compared to monolayer graphene. The review covers the physics of monolayer and bilayer graphene, including their behavior in an external magnetic field, the integer and fractional quantum Hall effects, and the role of electron correlations. It also discusses the transport properties of graphene in the absence of a magnetic field, the influence of disorder, and the manipulation of band gaps and magnetic properties through external interactions. The review highlights the differences and similarities between monolayer and bilayer graphene, focusing on thermodynamic properties such as compressibility, plasmon spectra, weak localization correction, and optical properties. Additionally, it explores the confinement of electrons in graphene nanoribbons and the effects of substrate interactions, adsorbed atoms, lattice defects, and doping on the band structure.The electronic properties of graphene, a two-dimensional crystal of carbon atoms, are unique and novel. Graphene exhibits massless chiral Dirac fermions, leading to interesting effects such as the quantum Hall effect and fractional quantum Hall effect. Bilayer graphene, composed of two stacked monolayers, has a quadratic low-energy band structure with different scattering properties compared to monolayer graphene. The review covers the physics of monolayer and bilayer graphene, including their behavior in an external magnetic field, the integer and fractional quantum Hall effects, and the role of electron correlations. It also discusses the transport properties of graphene in the absence of a magnetic field, the influence of disorder, and the manipulation of band gaps and magnetic properties through external interactions. The review highlights the differences and similarities between monolayer and bilayer graphene, focusing on thermodynamic properties such as compressibility, plasmon spectra, weak localization correction, and optical properties. Additionally, it explores the confinement of electrons in graphene nanoribbons and the effects of substrate interactions, adsorbed atoms, lattice defects, and doping on the band structure.