The known results about the rationality vs irrationality of smooth Fano complete intersections $X^n\subset\mathbb P^{n+c}$ of dimension $n=3,4,5$ and fixed type $(d_1,\ldots, d_c)$ suggest an uniform approach to treat several open cases: index one; index two; quartic fourfolds and fivefolds; etc. From one hand one would like to decide the rationality/irrationality of every element in the numerous cases where the stable irrationality of the very general element is known (e.g. quartic fourfolds and fivefolds, quintic fivefolds, etc); from the other hand one hopes to put some further light on several longstanding conjectures (e.g. the irrationality of the very general cubic fourfold). After an introduction of the general problem and after recalling the state of the art, we shall present some of our recent results on these topics.

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