Some problems in algebraic group theory

For smooth actions of compact Lie groups on differentiable manifolds, the existence of a smooth slice transversal to each orbit gives a clear description of the local structure. In 1973, D. Luna proved the existence of a slice in the etale topology at a closed orbit, for reductive algebraic groups acting on an affine variety, over an algebraically closed field of characteristic zero. This thesis explores the extent to which Luna's methods work over an arbitrary field. Conditions for the quotient of a morphism to be etale are given, necessary and sufficient conditions are given for the existence of a slice on a smooth affine scheme, and a new proof is given of the isomorphism of the unipotent variety of a split connected, simple, semisimple algebraic group with the nilpotent variety of its Lie algebra.