Let $\varphi$ be an automorphism of the unit disc $\mathbb{D}$ and let $W_{\varphi}^{\alpha}$ be the weighted composition operator, acting on a Banach space of analytic functions in the unit disc, defined by $W_{\varphi}^{\alpha} f= (f \circ \varphi)(\varphi')^{\alpha}$ with $\alpha>0$. We observe that many common examples of Banach spaces of analytic functions like Korenblum growth classes, Hardy spaces, standard weighted Bergman and certain Besov spaces are invariant under these operators. Thus, the main goal of this talk will be to present a general approach to the spaces that satisfy this weighted conformal invariance property. Among other things, we will identify the largest and the smallest as well as the "unique" Hilbert space satisfying this property for a given $\alpha>0$. These results are part of a joint work together with Professor Alexandru Aleman.