  We review recent work on superlattices in monolayer and bilayer graphene. We
highlight the role of the quasiparticle chirality in generating new Dirac
fermion modes with tunable anisotropic velocities in one dimensional (1D)
superlattices in both monolayer and bilayer graphene. We discuss the structure
of the Landau levels and magnetotransport in such superlattices over a wide
range of perpendicular (orbital) magnetic fields. In monolayer graphene, we
show that an orbital magnetic field can reverse the anisotropy of the transport
imposed by the superlattice potential, suggesting possible switching-type
device applications. We also consider topological modes localized at a kink in
an electric field applied perpendicular to bilayer graphene, and show how
interactions convert these modes into a two-band Luttinger liquid with tunable
Luttinger parameters. The band structures of electric field superlattices in
bilayer graphene (with or without a magnetic field) are shown to arise
naturally from a coupled array of such topological modes. We briefly review
some bandstructure results for 2D superlattices. We conclude with a discussion
of recent tunneling and transport experiments and point out open issues.
