Accurate finite volume solution algorithms for the two dimensional Navier Stokes equations and the three dimensional Euler equations for both structured and unstructured grid topologies are presented. Results for two dimensional quadrilateral and triangular elements and three dimensional tetrahedral elements will be provided. Fundamental to the solution algorithm is a technique for generating multidimensional polynomials which model the spatial variation of the flow variables. Cell averaged data is used to reconstruct pointwise distributions of the dependent variables. The reconstruction errors are evaluated on triangular meshes. The implementation of the algorithm is unique in that three reconstructions are performed for each cell face in the domain. Two of the reconstructions are used to evaluate the inviscid fluxes and correspond to the right and left interface states needed for the solution of a Riemann problem. The third reconstruction is used to evaluate the viscous fluxes. The gradient terms that appear in the viscous fluxes are formed by simply differentiating the polynomial. By selecting the appropriate cell control volumes, centered, upwind and upwind-biased stencils are possible. Numerical calculations in two dimensions include solutions to elliptic boundary value problems, Ringleb’s flow, an inviscid shock reflection, a flat plate boundary layer, and a shock induced separation over a flat plate. Three dimensional results include the ONERA M6 wing. All of the unstructured grids were generated using an advancing front mesh generation procedure. Modifications to the three dimensional grid generator were necessary to discretize the surface grids for bodies with high curvature. In addition, mesh refinement algorithms were implemented to improve the surface grid integrity. Examples studied include a Glasair fuselage, High Speed Civil Transport, and the ONERA M6 wing. The role of reconstruction as applied to adaptive remeshing is discussed and a new first order error estimator is presented. Numerical examples of the remeshing procedure include both smooth and discontinuous flows. / Ph. D.