This paper presents a theoretical framework for understand ing plasma turbulence in astrophysical plasmas. It is motivated by observations of electromagnetic and densit y fluctuations in the solar wind, interstellar medium and galaxy clusters, as well as by models of particle heating in accretion disks. All of these plasmas and many others have turbulent motions at weakly collisional an d collisionless scales. This paper focuses on turbulence in a strong mean magnetic field (the guide field). T he key assumptions behind the theory developed here are that the turbulent fluctuations are anisotropic wit h respect to the mean field and that their frequency is low compared to the ion cyclotron frequency. The turbulen ce is assumed to be stirred (forced) at some system-specific outer scale L. The energy injected at this scale has to be dissipated into h eat, which ultimately cannot be accomplished without collisions. A kinetic cascadedevelops that brings the energy to collisional scales both in space and velocity. The nature of the kinetic c as ade in various scale ranges depends on the physics of plasma fluctuations that can exist there. There ar four special scales that separate physically distinct regimes: the electron gyroscale ρe, the ion gyroscaleρi , the mean free path λmfpi and the electron heat diffusion scale (mi/me)λmfpi (me andmi are electron and ion masses). In each of the scale ranges sepa rated by these scales, a number of physically meaningful and rigorously ju stifiable simplifications of the fully kinetic plasma description are possible. These are derived systematicall y vi a hierarchy of asymptotic expansions. The result is that, in each scale range, the fully kinetic proble m is reduced to a more physically transparent and computationally tractable system of equations, which are d erived in a rigorous way. In the “inertial range” above the ion gyroscale, the kinetic cascade separates into two parts: a cascade of Alfvénic fluctuations and a passive cascade of density and magnetic-field-strength fluc tuations. The former are governed by two fluid-like Reduced Magnetohydrodynamic (RMHD) equations at both the c ollisional and collisionless scales; the latter obey a linear kinetic equation along the (moving) field lines as ociated with the Alfvénic component (in the collisional limit, these passive fluctuations become the sl ow and entropy modes of the conventional MHD). In the “dissipation range” between the ion and electron gyroscales, there are again two cascades: the kineticAlfvén-wave (KAW) cascade governed by two fluid-like Electr on Reduced Magnetohydrodynamic (ERMHD) equations and a passive cascade of ion entropy fluctuations b oth in space and velocity. The latter cascade brings the energy of the inertial-range fluctuations that was dampe d by collisionless wave-particle interaction at the ion gyroscle to collisional scales and leads to ion heating. The KAW energy is similarly damped at the electron gyroscale and is converted into electron heat. The relation ship between the theoretical models proposed in this paper and astrophysical applications and observations is d i cussed in detail. Subject headings: magnetic fields—methods: analytical—MHD—plasmas—turbul ence