About analytic continuation of quantum field theories in non-integer dimensions

Analytic continuation in dimension has been used first as a way to regularize perturbative quantum field theory. But since the work of Wilson and Fisher, quantum field theory in d-dimension has been used more radically, to connect theories living say, in d=4, to theories in d=3 and d=2. Mathematically it's not fully clear what this means. I will give some thoughts about this subject, and I will describe some recent paradoxes which arise when one consider expansion of O(N) models around d=2, based on recent work with Fabiana De Cesare.