We study the approximation of the spectrum of a second-order elliptic differential operator by the Hybrid High-Order (HHO) method. The HHO method is formulated using cell and face unknowns which are polynomials of some degree k ≥ 0. Only cell unknowns correspond to eigenvalues and eigenfunctions, while the face unknowns help deliver superconvergence. Using the abstract theory of spectral approximation of compact operators in Hilbert spaces, we prove that the eigenvalues (super-)converge at rate (2k + 2) and the eigenfunctions at rate (k + 1) in the H1-seminorm. Our theoretical findings are verified on various numerical examples including smooth and non-smooth eigenfunctions. Moreover, we observe numerically in one dimension a convergence rate of (2k +4) for the eigenvalues for a specific value of the stabilization parameter. Mathematics Subjects Classification: 65N15, 65N30, 65N35, 35J05