The quantum description of electrons in graphene is governed
by a massless Dirac equation, with a particle speed about 1/300
that of light. In this circumstance relativistic effects become much
easier to observe than for electrons in a vacuum.
In this talk I will discuss some of these effects and show how they lead to
unique electron transport behaviors. Among these I will discuss the
Klein paradox, a transmission of electric current through barriers, and how it can
be exploited to create multiple species of massless particles, some
of which move through the system as if a magnetic field is present
even when none is applied. I will also discuss the analog of
"Coulomb implosion" -- a phenomenon which in real relativistic
quantum mechanics limits the charge of a nucleus -- and how this
may be related to the spontaneous generation of mass in an
interacting theory of electrons in graphene.