This paper deals with intertwining : this is a surgery which combines the dynamics of different complex polynomials into a single one of larger degree. We give a precise definition of it and study its geometric and topological properties. We first prove its existence in many cases by surgery. For these cases, it is proved that the definition yields a unique map up to conformal conjugacy. We then study its topological properties through a topological model (conjugate to its conformal counterpart if it exists). Afterwards, we provide a partial answer to when a polynomial can be reduced to the intertwining of polynomials. We are also capable, using Mãné-Sad-Sullivan theory, to show that intertwining depends continuously from the former polynomials, with some extra assumptions. As an application, we study the degree three connectedness locus, and especially the family having a fixed critical point. In the appendix we study a loose definition of polynomial-like mappings which involves non-quasiconformal maps.