Focusing of polarized laser beams gives rise to three-dimensional vector fields in the focal volume. This capability can be fully utilized through the use of high-order vector beams, e.g., radially and azimuthally polarized beams. We show that this provides completely new capabilities in the characterization of individual nano-objects by nonlinear optical microscopy. In particular, we use second-harmonic generation with vectorial excitation fields to probe the chirality of nano-objects and the morphology of metal nanoparticles. In addition, we probe the coupling of light to metal nanoparticles of low symmetry and to semiconductor nanowires. Finally, we design metal oligomers with structure matched to the focused vectorial field distribution.