The desire to characterize and model the atmospheres of the many extrasolar planets that have been discovered over the last three decades is a major driver of current astronomy. However, this goal is impacted by the lack of spectroscopic data on the molecules in question. As most atmospheres that can be studied are hot, some surprisingly so, this activity requires spectroscopic information not readily available from laboratory studies. This article will review the current status of available molecular spectroscopic data, usually presented as line lists, for studies of exoplanet atmospheres and, indeed, the atmospheres of other astronomical objects hotter than the Earth such as brown dwarfs, cool stars and even sunspots. Analysis of exoplanet transit spectra and the calculation of the relevant opacities often require huge datasets comprising billions of individual spectroscopic transitions. Conversely, the newly-developed high-resolution Doppler-shift spectroscopy technique has proved to be a powerful tool for detecting molecular species in exoplanet atmospheres, but relies on the use of smaller, highly accurate line lists. Methods of resolving issues arising from the competing demands of completeness versus accuracy for line lists are discussed.