In this paper we investigate the magnetohydrodynamic slip flow of an electrically conducting, viscoelastic fluid past a stretching surface. The main concern is to analytically investigate the structure of the solutions and determine the thresholds beyond which multiple solutions exist or the physical pure exponential type solution ceases to exist. In the case of the presence of multiple solutions, closed-form formulae for the boundary layer equations of the flow are presented for two classes of viscoelastic fluid, namely, the second-grade and Walter's liquid B fluids. Heat transfer analysis is also carried out for two general types of boundary heating processes, either by a prescribed quadratic power-law surface temperature or by a prescribed quadratic power-law surface heat flux. The flow field is affected by the presence of physical parameters, such as slip, viscoelasticity, magnetic and suction/injection parameters, whereas the temperature field is additionally affected by thermal radiation, heat source/sink, Prandtl and Eckert numbers. The regions of existence or non-existence of unique/multiple solutions sketched by the combination of these parameters are initially worked out by providing critical values and then velocity/temperature profiles and skin friction, coefficient/Nusselt number are examined and discussed. (C) 2011 Elsevier Masson SAS. All rights reserved.