Abstract. The notion of global conformal invariance>/i> (GCI) in Minkowski space allows to prove rationality of correlation functions and to extend the concept of vertex algebra to any number D of space-time dimensions. The case of even D, which includes a conformal stress-energy tensor with a rational 3-point function, is of particular interest. Recent progress, reviewed in the talk, includes a full account of Wightman positivity at the 4-point level for D=4, and a study of modular properties of thermal expectation values of the conformal energy operator.