The study of minimal Cantor systems and zero entropy dynamical systems provided recently striking results. Topological full groups of minimal subshifts provide finitely generated groups with original properties: they are simple, amenable, may have intermediate growth for some zero entropy subshifts. Frantzikinakis-Host proved the Sarnak conjecture for the logarithmic average and zero entropy dynamical systems with at most countably many invariant measures. Adamczewski-Bugeaud constructed transcendantal numbers from zero entropy subshifts. Hence a deep understanding of zero entropy systems is of particular importance by itself and for other topics like number, group theory but also for applications to quasicristallography, computer science or statistical physics. Despite substantial efforts to understand zero entropy and although many families are well understood few general results have been obtained. We aim to unify parts of existing results and to go deeper into zero entropy.