The little n-discs operad E_n controls the structure of an n-iterated monoid. It is natural to ask what are deformations of this structure or more generally what are the m-monoidal deformations of an n-iterated monoidal structure, m<=n. The first applications of the deformation theory of little discs implicitly appeared in physics: Drinfel'd's associators in connection with quantum groups and deformation quantization - both these supposedly unrelated theory of deformations are in fact controlled by the homotopy automorphisms of E_2 (Kontsevich, Tamarkin, Willwacher, Fresse). Another important application is in the study of embedding spaces. The crucial result about little discs operads is the fact that the inclusion of operads E_m -> E_n is formal over reals if and only if the codimension n-m is different from 1. It will be shown how this fact affects the relative deformation theory of little discs. (joint with T. Willacher)