When quantum fields are represented as operators on a Hilbert space, their two-point distributions naturally give rise to distributions of positive type. A number of basic results on such distributions, especially for translation invariant two-point distributions, have been known for a long time. E.g., the Bochner-Schwartz Theorem fully characterises translation invariant distributions of positive type. In this talk I will present two apparently new results on distributions of positive type, one pertaining to pointwise products and the other to methods for cutting and pasting. Both results were motivated by physical questions and extend the toolbox of theoretical physics. I will present the results in a general mathematical context before discussing their applications to quantum energy inequalities and to separable states for a free scalar QFT. This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)