  Two-dimensional gyrokinetics is a simple paradigm for the study of kinetic
magnetised plasma turbulence. In this paper, we present a comprehensive
theoretical framework for this turbulence. We study both the inverse and direct
cascades (the `dual cascade'), driven by a homogeneous and isotropic random
forcing. The key characteristic length of gyrokinetics, the Larmor radius,
divides scales into two physically distinct ranges. For scales larger than the
Larmor radius, we derive the familiar Charney--Hasegawa--Mima (CHM) equation
from the gyrokinetic system, and explain its relationship to gyrokinetics. At
scales smaller than the Larmor radius, a dual cascade occurs in phase space
(two dimensions in position space plus one dimension in velocity space) via a
nonlinear phase-mixing process. We show that at these sub-Larmor scales, the
turbulence is self-similar and exhibits power law spectra in position and
velocity space. We propose a Hankel-transform formalism to characterise
velocity-space spectra. We derive the exact relations for third-order structure
functions, analogous to Kolmogorov's four-fifths and Yaglom's four-thirds laws
and valid at both long and short wavelengths. We show how the general
gyrokinetic invariants are related to the particular invariants that control
the dual cascade in the long- and short-wavelength limits. We describe the full
range of cascades from the fluid to the fully kinetic range.
