The functional renormalisation group its mathematics and applications to asymptotic safety We present a regularisation scheme for scalar Quantum Field theories that enables a flexible and mathematically consistent formulation of interacting theories in arbitrary dimensions. In contrast to a lattice approach it retains the smooth features of spacetime and the infinite de grees of freedom such that in particular the rotational symmetry can be left unbroken. In this framework we give a mathematically rigorous derivation of the Wetterich equation as well as sufficient conditions for the passage to the limit of vanishing regularisation. We also introduce an iterative construction procedure for exact solutions to the Wetterich equation that works by producing higher order correlation functions form the renormalisation group flow of lower order correlators. Then a generalisation of Quantum Electrodynamics is considered in the asymptotic safety framework and particular solutions are found that reproduce physical results in a low energy regime. Finally the applicability of the introduced regularisation scheme to the ϕ4 theory is proved. It follows from an integrability statement that can be thought of as a generalisation of Fernique’s theorem on exponential tails of Gaußian measures.