In this paper we introduce a general class of transformations of (all or most of) the class $\mathfrak{M}_{L}(\mathbb{R}^{d})$, of $d$-dimensional Lévy measures on $\mathbb{R}^{d}$, into itself. We refer to transformations of this type as $\Upsilon$ transformations (or Upsilon transformations). Closely associated to these are mappings of the set $\mathcal{ID}(\mathbb{R}^{d})$ of all infinitely divisible laws on $\mathbb{R}^{d}$ into itself. In considerable generality, the mappings are one-to-one, regularising and bi-continuous. Furthermore, in many cases the transformations have a stochastic interpretation in terms of stochastic integrals with respect to Lévy processes.