Abstract The behaviour of the stress tensor under conformal transformations of both flat and curved spaces is investigated for free theories in a classical background metric. In flat space ℝ d it is derived by the operator product expansion of two stress tensors. For Weyl transformations of curved manifolds it is given by the effective potential for the metric. In four dimensions the general form of the potential and its consistency conditions are analysed. These issues are relevant for the possible generalizations of the central charge in higher dimensions. The related subject of the Casimir effect is studied by means of closed expressions for the bosonic partition function on the manifoldsT d and S 1 ×S d −1 . The general relationship between the Casimir effect on ℝ×S d −1 and the trace anomaly is emphasized.