This is the first in a series of papers, the overall objective of which is the formulation of a new covariant approach to nonequilibrium statistical mechanics in classical general relativity. The object here is the development of a tractable theory for self-gravitating systems. It is argued that the “state” of an N-particle system may be characterized by an N-particle distribution function, defined in an 8 N-dimensional phase space, which satisfies a collection of N conservation equations. by mapping the true physics onto a fictitious “background” spacetime, which may be chosen to satisfy some “average” field equations, one then obtains a useful covariant notion of “evolution” in response to a fluctuating “gravitational force.” For many cases of practical interest, one may suppose (i) that these fluctuating forces satisfy linear field equations and (ii) that they may be modeled by a direct interaction. In this case, one can use a relativistic projection operator formalism to derive exact closed equations for the evolution of such objects as an appropriately defined reduced one-particle distribution function. By capturing, in a natural way, the notion of a dilute gas, or impulse, approximation, one is then led to a comparatively simple equation for the one-particle distribution. If, furthermore, one treats the effects of the fluctuating forces as “localized” in space and time, one obtains a tractable kinetic equation which reduces, in the newtonian limit, to the standard Landau equation.