Interstellar dust is responsible, through surface reactions, for the creation of molecular hydrogen, the main component of the interstellar clouds in which new stars form. Intermediate between small, gas-phase molecules and dust are the polycyclic aromatic hydrocarbons (PAHs). Such molecules could account for 2-30% of the carbon in the Galaxy, and may provide nucleation sites for the formation of carbonaceous dust. Although PAHs have been proposed as the sources of the unidentified infrared emission bands that are observed in the spectra of a variety of interstellar sources, the emission characteristics of such molecules are still poorly understood. Here we report laboratory emission spectra of several representative PAHs, obtained in conditions approximating those of the interstellar medium, and measured over the entire spectral region spanned by the unidentified infrared bands. We find that neutral PAHs of small and moderate size can at best make only a minor contribution to these emission bands. Cations of these molecules, as well as much larger PAHs and their cations, remain viable candidates for the sources of these bands.