The mechanisms driving the surfactant-enhanced spreading of droplets on the surface of solid substrates, and particularly those underlying the superspreading behaviour sometimes observed, are investigated theoretically. Lubrication theory for the droplet motion, together with advection–diffusion equations and chemical kinetic fluxes for the surfactant transport, leads to coupled evolution equations for the drop thickness, interfacial concentrations of surfactant monomers and bulk concentrations of monomers and micellar, or other, aggregates. The surfactant can be adsorbed on the substrate either directly from the bulk monomer concentrations or from the liquid–air interface through the contact line. An important feature of the spreading model developed here is the surfactant behaviour at the contact line; this is modelled using a constitutive relation, which is dependent on the local surfactant concentration. The evolution equations are solved numerically, using the finite-element method, and we present a thorough parametric analysis for cases of both insoluble and soluble surfactants with concentrations that can, in the latter case, exceed the critical micelle, or aggregate, concentration. The results show that basal adsorption of the surfactant plays a crucial role in the spreading process; the continuous removal of the surfactant that lies upon the liquid–air interface, due to the adsorption at the solid surface, is capable of inducing high Marangoni stresses, close to the droplet edge, driving very fast spreading. The droplet radius grows at a rate proportional to ta with a = 1 or even higher, which is close to the reported experimental values for superspreading. The spreading rates follow a non-monotonic variation with the initial surfactant concentration also in accordance with experimental observations. An accompanying feature is the formation of a rim at the leading edge of the droplet. In some cases, the drop spreads to form a ‘pancake’ or creates a ‘secondary’ front separated from the main droplet.