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The development of fast numerical methods for multilevel radiative transfer(RT) applications often leads to important breakthroughs in astrophysics,because they allow the investigation of problems that could not be properlytackled using the methods previously available. Probably, the most familiarexample is the so-called Multilevel Accelerated $\Lambda$-Iteration (MALI)technique of Rybicki & Hummer for the case of a local approximate operator,which is based on Jacobi iteration. However, there are superioroperator-splitting methods, based on Gauss-Seidel (GS) and SuccessiveOverrelaxation (SOR) iteration, which provide a dramatic increase in the speedwith which non-LTE multilevel transfer problems can be solved in one, two andthree-dimensional geometries. Such RT methods, which were introduced byTrujillo Bueno & Fabiani Bendicho ten years ago, are the main subject of thefirst part of this paper. We show in some detail how they can be applied forsolving multilevel RT problems in spherical geometry, for both atomic andmolecular line transitions. The second part of the article addresses the issueof the calculation of the molecular number densities when the approximation ofinstantaneous chemical equilibrium turns out to be inadequate, which happens tobe the case whenever the dynamical time scales of the astrophysical plasmaunder consideration are much shorter than the time needed by the molecules toform.Comment: 23 pages, 8 figures, to appear in the proceedings of GRETA 2005  "Radiative transfer and applications to very large telescopes" EAS  Publications series, EDP Sciences, Ed. Ph. Stee (in press

Very Efficient Methods for Multilevel Radiative Transfer in Atomic and Molecular Lines