Low-mass stars: Open problems all along their evolution

Although low-mass stars have a comparatively simple structure, current stellar models are far from reproducing them accurately. In light of the observational progress, in particular of asteroseismology, the deficits of the theory of stellar structure and evolution become increasingly evident, and can no longer be hidden under the carpet of observational errors. In my brief review I will discuss a number of obvious problems of the models, which are mainly – and not surprisingly so – connected with convective and other mixing processes. They begin already on the pre-main sequence and continue throughout the complete evolution. In addition to the deficits in the treatment of physical processes I will also address our difficulties in obtaining completely satisfying agreement between different numerical codes. It will be shown that this concerns already the main sequence evolution of low-mass stars without a convective core. Keeping this in mind, it is no surprise that models for stars on the Asymptotic Giant Branch differ widely.