A measure is said to be spectral if its -space has an orthonormal basis of exponential functions. A class of “fractal measures” discovered by Hutchinson in 1981 has been shown to contain many members which are spectral. More recently, it has been discovered that some of these measures in fact have infinitely many spectra, which relate in a surprisingly nice way. These numerous spectra give rise to unitary operators, which themselves have been shown to have interesting and fractal-like properties. In this talk, I will discuss these phenomena while giving details of my recent work in exploring them in the case of a relatively unexplored subclass of Hutchinson’s fractal measures.