Scale and conformal symmetries play a central role in various areas of theoretical physics, ranging from statistical mechanics and quantum field theory to string theory. A natural question is under what conditions scale invariance implies the larger symmetry of conformal invariance. In this talk, I will discuss non-perturbative arguments for this enhancement in two and four dimensions, highlighting the assumptions under which they hold. I will first explain how Zamolodchikov's $c$-theorem, under suitable assumptions, can be used to establish the enhancement of scale invariance to local conformal invariance in two dimensions. I will then turn to four dimensions and discuss the argument of Dymarsky, Komargodski, Schwimmer, and Theisen based on dilaton scattering amplitudes and unitarity. I will conclude by commenting on possible generalisations to other dimensions and, time permitting, recent progress on the question.
