Combination band and overtone transitions of polycyclic aromatic hydrocarbon (PAH) cations in the 2000–2900 cm-1 (5-3.5 µm) region are implicated as carriers of the ‘quasi-continuum’ observed in the near-infrared (IR) emission spectra of many astronomical objects. In neutral PAHs, the strongest absorption features are concentrated in the 700–900 cm-1 (14–11 µm) range, which are associated with CH out-of-plane bending motions. Upon ionization, this shifts to the 1000–1600 cm-1 (10–6 µm) range, where bands are associated with C=C stretches and CH in-plane bends. Anharmonicity is required to accurately characterize the IR absorption spectrum of PAHs, indicated herein by the ability to directly assign the bands in high-resolution experimental absorption spectra of neutral and cationic phenanthrene, pyrene, and pentacene. Neutral PAHs are indicated as the source of the strong 3.3 and 11.2 µm astronomical PAH features, while the broad features in the 6–10 µm region and the ‘quasi-continuum’ from 3.5–5 µm stem from PAH cations. This study reinforces the need for including anharmonicity in the computation of IR absorption and emission spectra of larger and more complex PAHs. This is particularly pertinent to the interpretation of data returned by JWST.