The Marangoni flows in a shallow cavity subject to uniform heat fluxes on all sides are investigated. A power law model is used to characterize the non-Newtonian fluid behavior of the fluid. The system with an underformable free upper surface is assumed to be under a zero gravity environment. The governing parameters for the problem are the thermal Marangoni number Ma, power-law index n, Prandtl number Pr and cavity aspect ratio aspect ratio A. An analytical solution, valid for an infinite layer (A >> 1), is derived on the basis of the parallel flow approximation. For the case of a layer heated from the bottom it is demonstrated that, for shear-thinning fluids (n < 1), the onset of convection is subcritical. For shear thickening fluids (n > 1), convection is found to occur at a supercritical Rayleigh equal to zero. For the case of a layer heated from all sides it is shown that multiple steady state solutions are possible, some of which are unstable. The effects of the non-Newtonian behavior on the fluid flow, temperature field and heat transfer are discussed. A good agreement is found between the analytical predictions and the numerical results obtained by solving the full governing equations. (c) 2012 Elsevier Ltd. All rights reserved.