The use of a combination of tunneling and optical spectroscopy to investigate the size and shape-dependent level structure and single-electron charging phenomena in semiconductor nanocrystals is reviewed. The artificial atom character of semiconductor nanocrystal quantum dots is manifested in both the discrete level structure and in the charging multiplicity of the single-electron tunneling data, revealing s and p atomic-like states. Such states can be directly imaged using scanning tunneling microscopy, providing the extent and symmetry of the envelope wavefunctions. A detailed description of the effect of the tunneling geometry on the single-electron tunneling spectra is presented. Correlation of the optical and tunneling data allows for the assignment of the level spectrum. The generality of this powerful combination is further demonstrated in the study of quantum rods that manifest the transition from zero-dimensional quantum dots to one-dimensional quantum wires.