We present results of Monte Carlo simulations of polymer brushes attached to a spherical surface. We have varied the chain length, surface radius, and the grafting density, and the analysis has included the size, shape, and orientation as well as segment density profiles. We also compared different methods of generating the starting configurations to establish the results being independent of the initial structure. For flexible molecules, the segment density profiles agree with earlier results from self-consistent field calculations, and there is no depletion zone for the chain ends close to the surface, except for the largest molecules. Increasing the persistence length, the orientation of the molecules increases and the chain density close to the surface decreases, but there is still a finite probability of finding a chain end close to the particle surface. We have also looked at the effect of restricting the available space for the chains by an outer bounding surface and the change in molecular shape and size is then seen much earlier compared to the effect on the subunit density profiles.