In this talk, we discuss how to interpret the critical gravitational collapse processes via renormalization group (RG). As an instructive example, we propose extremal black holes (BH) as critical points of a class of gravitational collapses, and find the transition between Continuous Self Similarity (CSS) and Discrete Self Similarity (DSS) behaviours as we tune the parameters of the extremal BH spacetime. In particular, we explicitly show that the DSS solution found here can be interpreted as RG limit cycles, and the transition between CSS and DSS regimes occurs as the stable and unstable fixed points collide and move to the complex plane. We argue that the DSS solutions found in spherically symmetric gravitational collapses can be similarly interpreted. It turns out that various phenomena in non-gravitational systems with RG limit cycles, including DSS correlation function, DSS scaling laws in correlation length and order parameters can also be identified in gravitational critical collapses.