  This article provides a pedagogical review on Klein tunneling in graphene,
i.e. the peculiar tunneling properties of two-dimensional massless Dirac
electrons. We consider two simple situations in detail: a massless Dirac
electron incident either on a potential step or on a potential barrier and use
elementary quantum wave mechanics to obtain the transmission probability. We
emphasize the connection to related phenomena in optics, such as the
Snell-Descartes law of refraction, total internal reflection, Fabry-P\'erot
resonances, negative refraction index materials (the so called meta-materials),
etc. We also stress that Klein tunneling is not a genuine quantum tunneling
effect as it does not necessarily involve passing through a classically
forbidden region via evanescent waves. A crucial role in Klein tunneling is
played by the conservation of (sublattice) pseudo-spin, which is discussed in
detail. A major consequence is the absence of backscattering at normal
incidence, of which we give a new shorten proof. The current experimental
status is also thoroughly reviewed. The appendix contains the discussion of a
one-dimensional toy model that clearly illustrates the difference in Klein
tunneling between mono- and bi-layer graphene.
