Self-similarity is a pervasive concept in mathematics, appearing in fields as diverse probability and operator theory, and mathematical physics. In this talk, we offer a novel and comprehensive perspective on the rich structure of self-similarity by exploring its connections with group representation theory and operator algebras. We will discuss the surprising roles of the Bessel operator and the Laplacian within this framework. By the end of the talk, we will uncover how these structures enhance our understanding of scaling limits and universality, with a particular focus on the spectrum of large random matrices.