An implicit space-marching, finite-difference procedure for solving the full Navier-Stokes equations was modified to handle unsteady flows with heat transfer. The compressible form of the equations are used on curvilinear, body-fitted coordinates. Primitive variables are used as the unknowns. The scheme uses a novel single-sweep, parabolized pressure Poisson solver to correct the pressure after each global, Navier-Stokes sweep;Only external flow results are presented in this report. Both flow and heat transfer results are presented for steady flow over a semi-infinite flat plate. Similar results are shown for an impulsively started flat plate. Flow with heat transfer over a cylinder in crossflow is also computed. Results for steady flow at a Reynolds number of 40 are shown first. Flows and heat transfer for the impulsively started cylinder at a Reynolds number of 550 validate the extension of the code to unsteady flows. The heat transfer results for this last flow are believed to be the first such results reported;The pressure backsweep is found to be a stabilizing element in the code for this geometry, as well as a method to accelerate convergence;The method was found to be accurate and fast for steady flows. Unsteady flows, in the author's opinion, were not as satisfactorily handled.