The application of an impinging sweeping jet, which oscillates periodically with a large angle, to convective heat transfer has received attention owing to its capability to provide a more spatially uniform and enhanced heat removal rate when compared to a steady jet. Herein, we study how the surface curvature affects the heat transfer performance of a sweeping jet and couple it with the representative flow characteristics. Heat transfer measurement and quantitative flow visualization are conducted experimentally for concave and convex surfaces as well as a flat surface. Whereas concave surfaces have a better heat transfer rate than a flat surface, the enhancement of the heat transfer is relatively small for a convex surface. For both concave and convex surfaces, the Nusselt number does not increase monotonically with the curvature magnitude but has a peak for a moderate curvature. The variation in heat transfer performance with the surface curvature is correlated with the phase-averaged velocity profile of the wall jet deflected after an impingement and the turbulence kinetic energy inside the jet. For both concave and convex surfaces, the wall jet becomes thinner than a flat surface in general, which contributes to improved heat transfer. However, whereas the turbulence kinetic energy is significantly larger for a concave surface of a moderate curvature than that of a flat surface, the turbulence kinetic energy for a convex surface is reduced from that of a flat surface, resulting in degradation of the heat transfer performance.