A noncanonical Hamiltonian theory of dynamical systems is presented and applied to magnetic field line flow. The theory allows all of the theorems of Hamiltonian mechanics (most importantly, Noether's theorem, relating symmetries and invariants) to be applied to the magnetic field line system. The theory is not restricted to any particular geometry. An elementary derivation of noncanonical Hamiltonian perturbation theory, based on Lie transforms, is also presented. As an example, the perturbation theory is applied to magnetic field line flow in nearly azimuthally symmetric geometry. Other applications are to the adiabatic motion of charged particles.