We investigate the flow past circular, square, and triangular cylinders embedded on spherical and cylindrical surfaces and quantify the effect of the embedding surface curvature on the flow characteristics. For the circular cylinder, we examine three Reynolds numbers Re=40,100, and 1000 based on cylinder diameter (D), and vary the normalized embedding surface curvature values κD, where κ is the maximum principal curvature, from 0 (flat surface) to 1/12. Measures of fluid dynamical quantities of interest: drag coefficient Cd, lift coefficient Cl, Strouhal number St, pressure coefficient (Cp), friction coefficient (Cf), vorticity magnitude (ω∗), street speed Vst, street wavelength λ, length of stationary vortex pair (L), and separation angle (θsep) show an insignificant effect of the surface curvature. For the range of embedding surface curvature values, the dynamics of the flow past a stationary circular cylinder exhibits near universality, and is essentially independent of the embedding surface curvature. Our conclusion is further consolidated by the similar insensitivity to the embedding surface curvature exhibited by flows past a stationary cylinder having a square or triangular cross-section.