Universality broadly construed is well understood since the 70s. Particular universality classes are newer and will likely continue to be discovered, but they all come to be in a qualitatively similar way.

If anyone has genuine interest, this review from shortly after the clarifying development of renormalization group theory might be a nice place to start: https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.46....