Criteria for the density property of complex manifolds

In this paper we suggest new effective criteria for the density property. This enables us to give a trivial proof of the original Anders\'en-Lempert result and to establish (almost free of charge) the algebraic density property for all linear algebraic groups whose connected components are different from tori or $\C_+$. As another application of this approach we tackle the question (asked among others by F. Forstneri\v{c}) about the density of algebraic vector fields on Euclidean space vanishing on a codimension 2 subvariety.