Dilation and conformal invariance in quantum field theory require the existence of a renormalization group fixed point and exploit partial summation of the perturbation series. I an early work of G. Mack and the speaker a skeleton expansion is constructed (in terms of conformally invariant dressed propagators and vertex functions) that is free of ultraviolet divergences in a certain range of anomalous dimensions. In a second part of the lecture the (stronger) condition of global conformal invariance (GCI) is explored that yields rational correlation functions and a higher (in particular D = 4) extension of the notion of vertex algebra. Copnformal partial wave expansions are used in a recent work of N. Nikolov, K.-H. Rehren and the author to establish Wightman positivity at the 4-point level.