Research concerning Gagliardo-Nirenberg inequality involves a general estimate
$$A(D^k f)\lesssim B(D^j f) C(f),\text{ for } k\leq j, f\in W^{j,0}.$$
The original versions from 1959 consider $A, B,$ and $C$ to be the powers of Lebesgue norms and $f$ to be the Sobolev function defined either on the whole space or on a Lipschitz domain. Much generalisation and refinements were presented, in the talk, we mention some of them both known and unpublished including the choice of spaces generating functionals $A, B,$ and $C$, methods of verifying the optimality, cases of fractional derivatives and possible nonlinearity on the right-hand side.