In this work, we develop a spectral theory for hypergraph limits. We prove the convergence ofthe spectra of adjacency and Laplacian matrices for hypergraph sequences converging in the 1-cutmetric. On the other hand, we give examples of matrix operators associated with hypergraphswhose spectra are not continuous with respect to the 1-cut metric. Furthermore, we show thatthese operators are continuous with respect to other cut norms.