  Hamiltonian structure is pursued and uncovered in collisional and
collisionless gyrokinetic theory. A new Hamiltonian formulation of
collisionless electromagnetic theory is presented that is ideally suited to
implementation on modern supercomputers. The method used to uncover this
structure is described in detail and applied to a number of examples, where
several well-known plasma models are endowed with a Hamiltonian structure for
the first time. The first energy- and momentum-conserving formulation of full-F
collisional gyrokinetics is presented. In an effort to understand the
theoretical underpinnings of this result at a deeper level, a \emph{stochastic}
Hamiltonian modeling approach is presented and applied to pitch angle
scattering. Interestingly, the collision operator produced by the Hamiltonian
approach is equal to the Lorentz operator plus higher-order terms, but does not
exactly conserve energy. Conversely, the classical Lorentz collision operator
is provably not Hamiltonian in the stochastic sense.
