Graphene, a two-dimensional allotrope of carbon, has drawn much attention since its first experimental isolation. Much of this fascination stems from its exotic low-energy dynamics that is governed by the ``Dirac equation,’’ a quantum mechanical law of motion that was originally discovered for relativistic particles such as neutrinos. Unlike for real neutrinos, however, the Dirac equation for electrons in graphene is generically massless. In this talk I discuss how a mass term can be induced in the material and what the consequences of such mass are. Also unlike the situation for real, relativistic particles, the mass for the charge carriers in graphene can be space-dependent. I will discuss the manifestations of this exotic property for the electrical characteristics of the material. I will discuss evidence of the discussed effects in scanning tunneling microscopy measurements.