Time-dependent density functional theory (TDDFT) and its modification, the Tamm-Dancoff approximation to TDDFT, are employed to calculate the electronic excitation energies and oscillator strengths for a series of polycyclic aromatic hydrocarbon radical cations. For the radical cations of naphthalene and anthracene, TDDFT using the Becke-Lee-Yang-Parr functional and the 6-31G** basis set provides the excitation energies that are roughly within 0.3 eV of the experimental data. The assignments of the electron transitions proposed by TDDFT accord with the previous assignments made by accurate ab initio calculations, except that TDDFT indicates the existence of a few additional transitions of pi*<--sigma character among the several low-lying transitions. The calculated energies for these pi*<--sigma transitions are found to be consistent with the onset of a sigma electron ionization manifold in the photoelectron spectra. For the pyrene radical cation, TDDFT supports the previous assignments made by semiempirical calculations, whereas for the perylene radical cation, TDDFT suggests the energy ordering of the three lowest-lying excited states be changed from those of the semiempirical results.