Recent Progress in Asymptotic Safety

Free and interacting fixed points of the renormalization group play an important role in particle and statistical physics. High energy fixed points such as in asymptotic freedom or asymptotic safety are crucial for a fundamental definition of quantum field theory. Low energy fixed points relate to (quantum) phase transitions and critical phenomena. In this contribution, we explain the arising of weakly interacting fixed points in general 4d QFTs. This covers necessary and sufficient conditions for existence, and explicit examples. In addition, we also review the last progress in calculation of beta functions, anomalous dimensions up to four loop in the gauge and three loop in the Yukawa and scalar couplings. Moreover, we will overview the status of functional renormalization group and its implications for gravity.