We have modeled the family of interstellar IR emission bands at 3.3, 6.2, 7.7, 8.6, 11.3, and 12.7 mum by calculating the fluorescence from a size distribution of interstellar polycyclic aromatic hydrocarbons (PAHs) embedded in the radiation field of a hot star. It is found that the various emission bands are dominated by distinctly different PAHs, from molecules with much less than approximately 80 C atoms for the 3.3 mum feature, to molecules with 10(2)-10(5) C atoms for the emission in the IRAS 12 and 25 mum bands. We quantitatively describe the influence on the emergent spectrum of various PAH properties such as the molecular structure, the amount of dehydrogenation, the intrinsic strength of the IR active modes, and the size distribution. Comparing our model results to the emission spectrum from the Orion Bar region, we conclude that interstellar PAHs are likely fully, or almost fully, hydrogenated. Moreover, it is found that the intrinsic strengths of the 6.2 and 7.7 mum C-C stretching modes, and the 8.6 mum C-H in-plane bending mode are 2-6 times larger than measured for neutral PAHs in the laboratory. This difference is tentatively ascribed to interstellar PAHs being ionized, or, alternatively, to their being highly asymmetric. It is furthermore concluded that the earlier assignment of the 3.4 mum subfeature to the hot band of the aromatic C-H stretching mode is only marginally consistent with the available observational data. In some cases an additional component appears to be required to account for its intensity. The hot band may be responsible for a weaker shoulder in some cases. An assignment to the C-H stretching feature from ''superhydrogenated'' PAHs, i.e., PAHs with extra H atoms attached to the peripheral carbon atoms, is considered as the additional contributor. Finally, it is pointed out how future high-quality observations of the emission from interstellar PAHs covering a large wavelength range, together with modeling of the type presented in the paper, could be used to study a number of important questions, such the preferred molecular structures of interstellar PAHs and their evolution in the interstellar medium.