In collaboration with the Applied Numerical Algorithms Group at Lawrence Berkeley National Laboratory, CASC scientists are developing high-resolution methods for solving continuum kinetic systems in complex phase space geometries. The motivating applications include the solution of kinetic models of fusion edge plasmas, wherein the preponderant magnetic field requires the efficient discretization of a complicated four-, five-, or six-dimensional phase space. The techniques being investigated include conservative, high-order finite volume methods in the method-of-lines formulation for advection problems that are coupled to implicit solvers for the field equations. These methods are being developed for mapped multiblock grids that enable alignment of the grid coordinate directions to accommodate strong anisotropy. Although motivated by the edge plasma application, the underlining ideas area being formulated and developed as generally as possible for broader application. This project is funded by the DOE Office of Science, Advanced Scientific Computing Research Applied Mathematics base research program.