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The photospheres of low-mass red giants show CNO isotopic abundances that arenot satisfactorily accounted for by canonical stellar models. The same is truefor the measurements of these isotopes and of the $^{26}$Al/$^{27}$Al ratio inpresolar grains of circumstellar origin. Non-convective mixing, occurringduring both Red Giant Branch (RGB) and Asymptotic Giant Branch (AGB) stages isthe explanation commonly invoked to account for the above evidence. Recently,the need for such mixing phenomena on the AGB was questioned, and chemicalanomalies usually attributed to them were suggested to be formed in earlierphases. We have therefore re-calculated extra-mixing effects in low mass starsfor both the RGB and AGB stages, in order to verify the above claims. Ourresults contradict them; we actually confirm that slow transport below theconvective envelope occurs also on the AGB. This is required primarily by theoxygen isotopic mix and the $^{26}$Al content of presolar oxide grains. Otherpieces of evidence exist, in particular from the isotopic ratios of carbonstars of type N, or C(N), in the Galaxy and in the LMC, as well as of SiCgrains of AGB origin. We further show that, when extra-mixing occurs in the RGBphases of population I stars above about 1.2 $M_{\odot}$, this consumes $^3$Hein the envelope, probably preventing the occurrence of thermohaline diffusionon the AGB. Therefore, we argue that other extra-mixing mechanisms should beactive in those final evolutionary phases.

On the Need for Deep Mixing in AGB Stars of Low Mass