A new numerical procedure has been developed for the finite-volume solution of the Euler equations on unstructured, triangular meshes using a flux-difference split, upwind method. The procedure uses a dual mesh system to implement the finite-volume scheme using conserved variables stored at the vertices of the triangles. The vertices of the triangles are located approximately in the center of the dual mesh cells and conservation is enforced about each dual mesh finite-volume. Techniques are developed for implementing Roe's approximate Riemann solver on unstructured grids. Higher order accuracy is achieved by using MUSCL-differencing. MUSCL-differencing is implemented on an unstructured grid by interpolating the values stored at the vertices of triangular elements to find the value at the outermost point of the three-point MUSCL-differencing formula. Flow solutions are computed using a four-stage Runge-Kutta time integration. Convergence is accelerated using non-standard weighting of the Runge-Kutta stages, variable time-steps, residual smoothing and residual min- imization. Applications and comparisons with structured grid solutions are made for a supersonic shock reflection problem, the supersonic flow over a blunt body, flow through a simple wedge inlet, and several AGARD 07 working group transonic airfoils. In general, the solutions computed by the upwind solver on the unstructured grids were as accurate as upwind solutions on a structured mesh. The blunt body solution, and some of the transonic airfoil solutions on the unstructured meshes, appeared to be less accurate than the structured mesh solutions. Fortuitously, the structured meshes used in these solutions tended to line up with the shock waves present in the flow-field. The upwind flux-differencing scheme captured the shock wave with greatest accuracy when it was applied normal to the discontinuity, as was done in these test cases.