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This survey, which contains very few proofs, addresses the general question:Over a given type of field, is there a natural class of varieties whichautomatically have a rational point? Fields under consideration here include:finite fields, p-adic fields, function fields in one or two variables over analgebraically closed field. Classical answers are given by theChevalley-Warning theorem and by Tsen's theorem. More general answers wereprovided by a theorem of Graber, Harris and Starr and by a theorem of Esnault.The latter results apply to rationally connected varieties. We discuss thesevarieties from various angles : weak approximation, R-equivalence, Chow groupof zero-cycles. Ongoing work on `rationally simply connected' varieties overfunction fields in two variables is also mentioned. A common thread in thisreport is the study of the special fibre of a scheme over a discrete valuationring: if the generic fibre has a simple geometry, what does it imply for thespecial fibre?

Vari'et'es presque rationnelles, leurs points rationnels et leurs d'eg'en'erescences