Spectral graph theory is the study of the relationship between the graphical properties of a graph and the spectral properties (i.e., eigenvalues and eigenvectors) of various matrices associated with that graph, most commonly the adjacency matrix. The spectrum of the adjacency matrix of a cubic graph (i.e., one where each vertex has three neighbours) on n vertices is a set of n real numbers lying in the interval [-3,3] and it determines a surprising amount of information about the graph. A spectral gap set X is an open subset of (-3,3) with the property that there are an infinite number of cubic graphs whose spectrum is disjoint from X.  For example, the interval (-3,-2) is a spectral gap set because the infinite family of cubic line graphs has no eigenvalues in (-3,-2), and in fact the precise list of all cubic graphs whose spectrum avoids (-3,-2) is known.  Krystal Guo and Bojan Mohar showed that the interval (-1,1) is a spectral gap set for cubic graphs, and recently Alicia Kollár and Peter Sarnak demonstrated the same result for (-2,0) and in addition, showed that any spectral gap interval has length at most 2. In this talk I describe some recent work, joint with Krystal Guo, where we give exact characterisations of the cubic graphs with spectra avoiding (-1,1) and those with spectra avoiding (-2,0). These exact characterisations allow us to deduce that (-1,1) is a maximal spectral gap set, thereby answering a question of Kollár and Sarnak. The talk is largely non-technical and should be accessible to anyone familiar with basic graph theory and linear algebra.