The equations of quantum mechanics provide a complete framework to compute the spectra of molecules; doing this accurately requires detailed numerical treatment of both the electronic and nuclear motion. To obtain transition frequencies close to experimental accuracy requires the consideration of many effects not usually considered in standard electronic structure calculations; conversely experience shows that it is often possible to compute transition intensities with lower uncertainty than they can be measured. The seminar will discuss some of the methods used for these studies focusing on atmospherically and astronomically important molecules such as water, CO2 and H3+.