This work investigates three main topics. The first of these is the development and comparison of time integration schemes on two-dimensional unstructured meshes. Both explicit and implicit solution algorithms for the two-dimensional Euler equations on unstructured grids are presented. Cell-centered and cell-vertex finite volume upwind schemes utilizing Roe’s approximate Riemann solver are developed. For the cell-vertex scheme, a four stage Runge-Kutta time integration with and without implicit residual averaging, a point Jacobi method, a symmetric point Gauss-Seidel method, and two methods utilizing preconditioned sparse matrix solvers are investigated. For the cell-centered scheme, a Runge-Kutta scheme, an implicit tridiagonal relaxation scheme modeled after line Gauss-Seidel, a fully implicit LU decomposition, and a hybrid scheme utilizing both Runge-Kutta and LU methods are presented. A reverse Cuthill-McKee renumbering scheme is employed for the direct solver in order to decrease CPU time by reducing the fill of the Jacobian matrix. Comparisons are made for both first-order and higher-order accurate solutions using several different time integration algorithms. Higher-order accuracy is achieved by using multi-dimensional monotone linear reconstruction procedures. Results for flow over a transonic circular arc are compared for the various time integration methods. The second topic involves an interactive adaptive remeshing algorithm. The interactive adaptive remeshing algorithm utilizing a frontal grid generator is compared to a single grid calculation. Several device dependent interactive graphics interfaces have been developed along with a device independent DI-3000 interface which can be employed on any computer that has the supporting software including the Cray-2 supercomputers Voyager and Navier. Solutions for two-dimensional, inviscid flow over a transonic circular arc and a Mach 3.0 internal flow with an area change are examined. The final topic examined in this work is the capabilities developed for a structured three-dimensional code called GASP. The capabilities include: generalized chemistry and thermodynamic modeling, space marching, memory management through the use of binary C Input/Output, and algebraic and two-equation eddy viscosity turbulence modeling. Results are given for a Mach 1.7 three-dimensional analytic forebody, a Mach 1.38 axisymmetric nozzle with hydrogen-air combustion, a Mach 14.1 15° ramp, and Mach 0.3 viscous flow over a flat plate. The incorporation of these capabilities and the two-dimensional unstructured time integration schemes into a three-dimensional unstructured solver is also discussed. / Ph. D.