We have applied density functional theory in statistical mechanisms to study the heterogeneous nucleation of supersaturated vapors on spherical aerosol-like substrates. Our calculations reveal the inadequacy of the classical nucleation theory in describing the condensation of droplets on very small particles. The latter approach is particularly inaccurate both at high supersaturations and in the vicinity of the wetting transition. Comparisons are also made with recent experimental results for the condensation of supersaturated vapors on submicrometer particles.