Continuous phase transitions (critical points) of classical statistical physics systems with D spatial dimensions are often described by Euclidean conformal field theories. A necessary condition for this to happen is that the critical point should be invariant under spatial rotations and translations. However this condition is not sufficient: some exceptional critical points are described by scale invariant but not conformally invariant theories. If the classical statistical physics system is reflection positive, the corresponding Euclidean CFT describing its critical point will be unitary.