We present results from extensive Monte Carlo simulations of flexible and semiflexible excluded-volume polymer chains confined to impenetrable spherical surfaces. Our results are compared with the theoretical predictions for ideal chains by Mondescu and Muthukumar (MM) (Phys. Rev. E 1998, 57, 4411) and Spakowitz and Wang (SW) (Phys. Rev. Lett. 2003, 91, 166102), respectively. The SW prediction is found to be in better agreement with our simulation results than the MM prediction in all the cases studied. Conformation of chains of length L and persistence length l(p) restricted to move on a sphere of radius R can be reasonably described by the SW formalism in the regime L/(2pi) < R < 2l(p) For R/l(p) > 2, the mean square end-to-end distance, as a function of the chain length, evolves from a two-dimensional (2D) self-avoiding random walk behavior to a saturation value. A rigid rod behavior is recovered in the limit of short and stiff chains. Unlike ideal chains, excluded volume chains confined to a spherical surface of large enough radius display a transition from a disordered to an helicoidal state as chain stiffness is increased. We have characterized this transition through the bond orientational correlation function and the Monte Carlo results reflect a balance between the bending energy and the excluded volume interactions.