  We revisit the long-standing conjecture that in unitary field theories, scale
invariance implies conformality. We explain why the Zamolodchikov-Polchinski
proof in D=2 does not work in higher dimensions. We speculate which new ideas
might be helpful in a future proof. We also search for possible
counterexamples. We consider a general multi-field scalar-fermion theory with
quartic and Yukawa interactions. We show that there are no counterexamples
among fixed points of such models in 4-epsilon dimensions. We also discuss fake
counterexamples, which exist among theories without a stress tensor.
