Nonsimilar solution of the unsteady laminar incompressible magneto-hydrodynamic boundary layer flow and heat transfer for an electrically conducting fluid over two-dimensional and axisymmetric bodies in the presence of an applied magnetic field has been obtained. The effects of surface mass transfer, Joule heating and viscous dissipation are included in the analysis. Numerical computation have been carried out for the flow over a circular cylinder and a sphere using an implicit finite difference scheme in combination with a quasi-linearization technique. It is observed that magnetic field and suction cause the location of vanishing skin friction to move downstream while, the effect of injection is just the opposite. The effect of magnetic field on the skin friction is more pronounced as compared to its effect on the heat transfer. On the other hand, the heat transfer is strongly affected by the viscous dissipation and the effect is more for large times. However, heat transfer responds comparatively less to the fluctuations of the free stream than the skin friction. © Springer-Verlag 2001.