The authors of this paper have used the theory of Riordan matrices to introduce the notion of a Riordan graph in \cite{CJKM}. Riordan graphs are proved to have a number of interesting (fractal) properties, and they are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other families of graphs. The main focus in \cite{CJKM} is the study of structural properties of families of Riordan graphs obtained from certain infinite Riordan graphs. In this paper, we use a number of results in~\cite{CJKM} to study spectral properties of Riordan graphs. Our studies include, but are not limited to the spectral graph invariants for Riordan graphs such as the adjacency eigenvalues, (signless) Laplacian eigenvalues, nullity, positive and negative inertias, and rank. We also study determinants of Riordan graphs, in particular, giving results about determinants of Catalan graphs.