  Graphene, the first truly two-dimensional (one atom thin) material, possesses
strongly nonlinear electrodynamic and optical properties. At low (microwave,
terahertz) frequencies this results from the unique electronic property of
graphene - the "relativistic", linear energy dispersion of its electrons and
holes. At high (infrared, visible) frequencies its nonlinear response functions
have resonances due to the inter-band transitions between the electron and hole
bands. The position of these resonances on the frequency axis depends on the
Fermi energy, or the charge carrier density, which can be varied by applying a
gate voltage in the graphene field-effect transistor. This opens up the unique
opportunities to electrically control the nonlinear graphene response, - the
property which was not available in conventional (three-dimensional) nonlinear
optical materials. In this paper we give a short overview of the state of the
art in this rapidly developing area of physics and present some of our recent
results in this field. In particular, we discuss essential differences between
the nonlinear response functions and parameters of traditional
three-dimensional and new two-dimensional crystals.

Mesoscale and Nanoscale Physics