  We present a theoretical framework for describing electromagnetic kinetic
turbulence in a multi-species, magnetized, pressure-anisotropic plasma.
Turbulent fluctuations are assumed to be small compared to the mean field, to
be spatially anisotropic with respect to it, and to have frequencies small
compared to the ion cyclotron frequency. At scales above the ion Larmor radius,
the theory reduces to the pressure-anisotropic generalization of kinetic
reduced magnetohydrodynamics (KRMHD) formulated by Kunz et al. (2015). At
scales at and below the ion Larmor radius, three main objectives are achieved.
First, we analyse the linear response of the pressure-anisotropic gyrokinetic
system, and show it to be a generalisation of previously explored limits. The
effects of pressure anisotropy on the stability and collisionless damping of
Alfvenic and compressive fluctuations are highlighted, with attention paid to
the spectral location and width of the frequency jump that occurs as Alfven
waves transition into kinetic Alfven waves. Secondly, we derive and discuss a
general free-energy conservation law, which captures both the KRMHD free-energy
conservation at long wavelengths and dual cascades of kinetic Alfven waves and
ion entropy at sub-ion-Larmor scales. We show that non-Maxwellian features in
the distribution function change the amount of phase mixing and the efficiency
of magnetic stresses, and thus influence the partitioning of free energy
amongst the cascade channels. Thirdly, a simple model is used to show that
pressure anisotropy can cause large variations in the ion-to-electron heating
ratio due to the dissipation of Alfvenic turbulence. Our theory provides a
foundation for determining how pressure anisotropy affects the turbulent
fluctuation spectra, the differential heating of particle species, and the
ratio of parallel and perpendicular phase mixing in space and astrophysical
plasmas.
