We employ Monte Carlo simulations in the grand canonical ensemble (GCEMC) to investigate the impact of nonplanarity of a solid substrate on the locus of the prewetting phase transition. The substrate is modelled as a periodic sequence of furrows of depth D and periodicity s(x) in the x direction; the furrows are infinitely long in the y direction. Our results indicate that a necessary prerequisite for a prewetting transition is the formation of a(n approximately) planar interface between molecularly thin films and an adjacent (bulk) gas. Thus, in general the prewetting transition is shifted to larger chemical potentials because the formation of a planar film-gas interface is more difficult next to a nonplanar compared with a planar solid surface. However, this shift turns out to be nonmonotonic depending on D on account of subtle packing effects manifested in the deviation of the local density Deltarho(x,Deltaz;D) at the nonplanar solid surface from that at a planar substrate. If D becomes sufficiently large prewetting as a discontinuous phase transition is suppressed because inside the furrow a highly ordered film forms that prevents a planar film-gas interface from forming. (C) 2004 American Institute of Physics.