  This article provides a self-contained pedagogical introduction to the
relativistic kinetic theory of a dilute gas propagating on a curved spacetime
manifold (M,g) of arbitrary dimension. Special emphasis is made on geometric
aspects of the theory in order to achieve a formulation which is manifestly
covariant on the relativistic phase space. Whereas most previous work has
focussed on the tangent bundle formulation, here we work on the cotangent
bundle associated with (M,g) which is more naturally adapted to the Hamiltonian
framework of the theory. In the first part of this work we discuss the relevant
geometric structures of the cotangent bundle T*M, starting with the natural
symplectic form on T*M, the one-particle Hamiltonian and the Liouville vector
field, defined as the corresponding Hamiltonian vector field. Next, we discuss
the Sasaki metric on T*M and its most important properties, including the role
it plays for the physical interpretation of the one-particle distribution
function. In the second part of this work we describe the general relativistic
theory of a collisionless gas, starting with the derivation of the
collisionless Boltzmann equation for a neutral simple gas. Subsequently, the
description is generalized to a charged gas consisting of several species of
particles and the general relativistic Vlasov-Maxwell equations are derived for
this system. The last part of this work is devoted to a transparent derivation
of the collision term, leading to the general relativistic Boltzmann equation
on (M,g). The meaning of global and local equilibrium and the stringent
restrictions for the existence of the former on a curved spacetime are
discussed. We close this article with an application of our formalism to the
expansion of a homogeneous and isotropic universe filled with a collisional
simple gas and its behavior in the early and late epochs. [abbreviated version]
