Lanthanide-containing molecules pose significant challenge to quantum chemistry due to the small energy gap of the 4f, 5d, and 6s atomic orbitals of lanthanide that can produce a plethora of low-lying molecular excited states. The accurate description of these species by ab initio methods requires a comprehensive treatment of both electron correlation and relativistic effects, in particular the spin–orbit coupling (SOC) ones. Moreover, a high number of electronic excited states in a limited domain of energy can lead to breakdown of the Born-Oppenheimer (BO) approximation. Therefore, one must consider the possibility for states to be mixed by nonadiabatic and spin−orbit coupling simultaneously. This lecture provides an overview of recent advances in ab initio spectroscopy of lanthanide compounds. The issues addressed include the following items. 1) The development of a composite approach for accurately predicting spectroscopic and thermochemical properties of Ln-containing species. The major components of the composite protocol are the coupled cluster single, double, and perturbative triple CCSD(T) theory or multireference configuration interaction MRCI for situations demanding strongly multiconfiguration wave functions, corrections for outer-core−valence and higher order correlation, and SOC effects. The accuracy attainable within this approach is assessed by statistically calibrating the composite results for a large training set of Ln-containing molecules against experiment. 2) The development of first-principles approaches to accurately describing the infrared absorption spectra of open-shell Ln-containing polyatomic molecules beyond the BO approximation. We take here the example of the cerium triflouride molecule, CeF3, for which the spin-vibronic Hamiltonian and the diabatic dipole moment operator were constructed and parameterized in MRCI calculations via a hybrid approach employing quasi-diabatization technique. The Hamiltonian comprises all seven low-lying 4f1 electronic states coupled by six vibrational modes of the molecule, with the SOC being taken into account. The IR spectrum of CeF3 obtained from the variational solution of the Schrödinger equation features a complex structure owing to an intricate interplay of the Jahn–Teller, pseudo-Jahn–Teller, and spin-orbit coupling effects. Thus, the calculation results indicate the vibronic rather than vibrational origin of the spectrum under consideration, clearly showing the breakdown of the BO approximation for the Ln-containing species with partially filled 4f shell. This work was supported by the Russian Foundation for Basic Research (grant no. 13-03-01051) and the Ministry of Education and Science of the Russian Federation (project no. 4.3232.2017/4.6). V. G. Solomonik and A. N. Smirnov. J. Chem. Theory Comput. 2017. V. 13, P. 5240; O. A. Vasilyev, V. G. Solomonik. Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol. 2018, V. 61, N.3, P. 31