Exact solutions are presented of two-dimensional steady-state incompressible stagnation point flows at a current sheet separating two colliding plasmas. They describe the process of resistive field annihilation (zero reconnection) where the magnetic field in each plasma is strictly parallel to the current sheet, but may have different magnitudes and direction on its two sides. The flow in the (x, y) plane toward the current sheet, located at x = 0, may have an arbitrary angle of incidence and an arbitrary amount of divergence from or convergence towards the stagnation point. We find the most general form of the solution for the plasma velocity and for the magnetic field. For the z compenents of the flow and field, solutions in the form of truncating power series in y are found. The cases obtained in this study contain the solutions obtained by Parker, Sonnerup & Priest, Gratton et al. and Besser, Biernat & Rijnbeek as special cases. The role of viscosity in determining the flow and field configurations is examined. When the two colliding plasmas have the same viscosity and density, it is shown that viscous effects usually are important only in strongly divergent or convergent viscous flows with viscous Reynolds number of the order of unity or smaller. For astrophysical applications the viscous Reynolds number is usually high and the effects of viscosity on the interaction of plasmas of similar properties are small. The formulation of the stagnation-point flow problem involving plasmas of different properties is also presented. Sample cases of such flows are shown. Finally, a possible application of the results from this study to the earth's magnetopause is discussed briefly.