This paper investigates Marangoni boundary layer flow and heat transfer of copper-water nanofluid driven by exponential temperature with radiation effects. Unlike most classical works, five different types of nanoparticle shapes are taken into account, i.e. sphere, hexahedron, tetrahedron, column and lamina. The temperature distribution on the liquid-liquid or liquid-gas interface is closer to the realistic situation by assuming that it is an exponential function with the axis length. Similarity transformation is applied to reduce the governing nonlinear partial differential equations into a set of nonlinear ordinary differential equations, which are solved numerically by the shooting method coupled with Newton's scheme and Runge-Kutta algorithm. The effects of solid volume fraction, radiation parameter and empirical shape factor on the dimensionless velocity and temperature fields and the local Nusselt number are graphically illustrated and discussed. It is found that the solid volume fraction and nanoparticle shape have significant impacts on the thermal conductivity and sphere nanoparticle has better enhancement on heat transfer than other nanoparticle shapes. (C) 2016 Elsevier B.V. All rights reserved.