Electronic properties of materials are commonly described by quasiparticles that behave as non-relativistic electrons (or holes) with a finite mass and obey the Schrödinger equation. Here, we report a two-dimensional (2D) system with a dispersion relation that mimics relativistic (Dirac) particles with zero mass and an effective “velocity of light” c* ≈ 106 m/s. Our studies of graphene – a single atomic layer of graphite – have revealed a variety of unusual phenomena intrinsic to 2D Dirac fermions. In particular, we have observed that a) graphene’s conductivity never falls below a minimum value corresponding to the conductance quantum e2/h, even if concentrations of both electrons and holes tend to zero; b) the integer quantum Hall effect (QHE) in graphene is anomalous in that it occurs at half-integer filling factors; and c) the cyclotron mass m_c of massless charge carriers in graphene is described by Einstein’s equation E =m_c c*^2. The 2D system is not only interesting on its own but also allows one to access – in a condensed matter experiment – the subtle and rich physics of quantum electrodynamics and provides a bench-top setup for studies of phenomena relevant to cosmology and astrophysics.