  Magnetohydrodynamic (MHD) turbulence is encountered in a wide variety of
astrophysical plasmas, including accretion disks, the solar wind, and the
interstellar and intracluster medium. On small scales, this turbulence is often
expected to consist of highly anisotropic fluctuations with frequencies small
compared to the ion cyclotron frequency. For a number of applications, the
small scales are also collisionless, so a kinetic treatment of the turbulence
is necessary. We show that this anisotropic turbulence is well described by a
low frequency expansion of the kinetic theory called gyrokinetics. This paper
is the first in a series to examine turbulent astrophysical plasmas in the
gyrokinetic limit. We derive and explain the nonlinear gyrokinetic equations
and explore the linear properties of gyrokinetics as a prelude to nonlinear
simulations. The linear dispersion relation for gyrokinetics is obtained and
its solutions are compared to those of hot-plasma kinetic theory. These results
are used to validate the performance of the gyrokinetic simulation code {\tt
GS2} in the parameter regimes relevant for astrophysical plasmas. New results
on global energy conservation in gyrokinetics are also derived. We briefly
outline several of the problems to be addressed by future nonlinear
simulations, including particle heating by turbulence in hot accretion flows
and in the solar wind, the magnetic and electric field power spectra in the
solar wind, and the origin of small-scale density fluctuations in the
interstellar medium.
