Graphene, a single sheet of graphite, was first isolated in 2004 and since then it has been studied extensively,
The Dirac nature of its quasiparticle dynamics
gives rise to many unusual and exotic properties, such as an anomalous quantum Hall effect. It has been found that a bilayer of two graphene sheets also displays an unusual quantum Hall effect and, furthermore, when the bilayer is placed in a suitably configured field effect device, an asymmetry gap can be generated and the carrier concentration made different in each layer. This latter configuration provides a tunable semiconducting gap, which may give rise to important technological
applications in microelectronics. In this talk, I will introduce some of the basic ideas about graphene and discuss the frequency-dependent
optical conductivity of both monolayer and bilayer graphene, contrasting the two cases and comparing with experiment. The optical conductivity in the bilayer is found to have very rich structure and this can be understood from optical transitions in the band structure.