High-harmonic generation processes have been experimentally investigated in molecular hydrogen gas as both molecular dissociation and ionization occur in the gas. The high harmonics observed in our experiment can be mainly attributed to neutral hydrogen molecules. Besides the harmonic spectra, two strong Lyman lines from the dissociated hydrogen atoms and the emission from the neutral hydrogen molecules are observed. Several possible excitation and dissociation channels of hydrogen molecules are proposed. It is found that resonance effects are important for both high-harmonic generation and molecular dissociation. Theoretical calculations using the time-dependent Hartree-Fock (TDHF) method for H-2 reproduce some of the resonance effects.