Spectral graph theory is the discipline studying what properties of a graph can be deduced from the eigenvalues of associated matrices. One may wonder how far one can take this perspective. Is it typical for a random graph to be characterized up to isomorphism by spectral information? In this talk, I will discuss some of the history of this question as well as more recent progress. My main focus will be on a new line-of-attack that reduces the problem to studying random abstract-algebraic objects associated to the graph, such as groups and modules.