This lecture is devoted to a survey on explicit stability results in Gagliardo-Nirenberg-Sobolev and logarithmic Sobolev inequalities. Generalized entropy methods based on carré du champ computations and nonlinear diffusion flows can be used for proving inequalities in sharp form. Under restrictions on the functions, these methods also provide remainder terms which control various distances to the set of optimal functions. A result without restriction will also be presented for some Sobolev and logarithmic Sobolev inequalities.