Abstract. We give a snapshot of the rapidly developing field of ultracold polar molecules and walk the reader through the papers appearing in this Topical Issue.PACS. 33. Molecular properties and interactions with photons -33.80.Ps Optical cooling of molecules; trapping -34. Atomic and molecular collision processes and interactions -39. Instrumentation and techniques for atomic and molecular physics

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Numerical calculations of vibrational levels of alkali dimers close to the dissociation limit are developed in the framework of a Fourier Grid Hamiltonian method. The aim is to interpret photoassociation experiments in cold atom samples. In order to avoid the implementation of very large grids we propose a mapping procedure adapted to the asymptotic R Ϫn behavior of the long-range potentials. On a single electronic potential, this allows us to determine vibrational wave functions extending up to 500a 0 using a minimal number of grid points. Calculations with two electronic states, A 1 ⌺ u ϩ and b 3 ⌸ u states, both correlated to the Rb(5s)ϩRb(5 p) dissociation limit, coupled by fine structure are presented. We predict strong perturbation effects in the Rb 2 (0 u ϩ ͒ spectrum, manifested under the 5s, 5p 2 P 1/2 dissociation limit by an oscillatory behavior of the rotational constants.