A study was made of the (n, n), (n, n'), and (n, n'~,) reactions which occur when a natural rubidium scattering sample is exposed to mono-energetic neutron fluxes of energies ranging from 120 to 1910 keV. Total cross sections, elastic scattering angular distributions and excitation functions for inelastic scattering were measured with a neutron time-of-flight spectrometer. Results are compared with Hauser-Feshbach theory. A 40 cc Ge(Li) detector was used in the time-gated mode to measure ~-rays from the (n, n'~,) reaction. The latter measurements were used to refine and extend the energy level schemes derived from neutron spectroscopy. Several new levels were discovered in the low-lying (below 1900 keV) energy level spectra of 8SRb and 87Rb. Gamma decay schemes and branching ratios were determined for the low-lying levels of 85Rb and 87Rb.