A systematic investigation has been carried out of the vibrational spectra of the hydrogen peroxide crystal and its deuterated analogue, using both i.r. and Raman spectroscopy, over the range 4000-50 cm-1. Mixtures of the two isotopic species up to approximately 95 % deuteration were also studied to identify the fundamentals of the hybrid molecule, HDO2. Single crystals of hydrogen peroxide oriented along two of the three crystallographic axes were examined in the Raman effect with polarized laser light. The O---H stretching bands are remarkably narrow in the Raman spectra: [Delta][nu] () = 20 cm-1 at - 193[deg]C compared with about 80 cm-1 in the i.r. Nearly all the O---H stretching components predicted by the factor group analysis were observed but a satisfactory identification of all components of the OOH deformation modes could not be achieved. The O---O stretching frequency in D2O2 (872 cm-1) is slightly higher than in H2O2, (871 cm-1) contrary to expectations. The hydrogen bonds in the H2O2 crystal appear somewhat less strong (by about 10-15 %) than those in ice.A normal coordinate analysis of the unit cell modes proved to be of considerable value in the assignment of observed frequencies. The values of the eleven principal force constants and the twenty interaction constants used to fit the 59 assigned frequencies appear reasonable and are comparable with values found for other hydrogen bonded systems.