Knowledge of the chemical bonding of HfO and HfB ground and low-lying electronic states provides essential insights into a range of catalysts and materials that contain Hf-O or Hf-B moieties. Here, we carry out high level multi-reference configuration interaction theory and coupled cluster quantum calculations on these systems. We compute full potential energy curves, excitation energies, ionization energies, electronic configurations, and spectroscopic parameters with large quadruple-zeta and quintuple-zeta quality correlation consistent basis sets.