ABSTRACT We present the results of a three year project to monitor broad absorption-line QSOs (BALQSOs) with both broadband imaging (for continuum flux changes) and spectrophotometry (for broad absorption-line variations). About 10% of all optically selected QSOs show broad absorption-lines blueward of the high ionization resonance emission lines, indicating that gas is flowing outward from the QSO continuum source at velocities as high as 0.2c. Of 54 BALQSOs monitored with broadband imaging, ~33% showed significant time variability with deviations of 0.1 to 0.3 magnitudes. The time scale for variability was at least three months to a year in the QSO rest frame. Spectroscopic monitoring of 23 BALQSOs resulted in the detection of broad absorption-line time variations in 15 BALQSOs. Six of these cases showed small (marginally significant) changes, while four objects showed large BAL changes (±0.2 - 0.4 in normalized intensity). All of the BAL time variations appear to be changes in the residual intensity, or normalized flux, within the line, rather than variations in the velocity structure of the outflowing BAL gas. Our primary hypothesis is that a variable photoionizing continuum causes a change in the ionization levels, and thus a redistribution of the fractional abundances of the ions, resulting in the strengthening or weakening of the absorption lines. The BAL changes will appear nearly simultaneous with the continuum variations if the BAL variation mechanism travels in step with the continuum photons, and the mechanism induces rapid changes in the ionization levels of the gas. Therefore, unlike studies of broad emission-line variability in active galactic nuclei, BAL variability does not (for the simplest models) yield direct information about the size of the region. Our studies of BAL time variability have suggested that the velocity of the BAL gas varies monotonically with distance from the continuum source. With this assumption, BAL time variability provides the opportunity for detailed study of the ionization properties, column density, and elemental abundance of the BAL gas. In our data, we have found cases of BAL variability that appear to be correlated with a change in continuum flux. However, we have also discovered a few cases where the BALs varied, but no change was detected (to a limit of ~0.06 magnitudes) in the broadband flux. These results suggest that the ionizing flux (lambda ~100 - 400A) varies with greater amplitude and possibly in a non-synchronous manner with the observed continuum flux (λ ~2000A). For one object, we observed a large continuum level change (factor of 2) without a corresponding BAL change. However, this same QSO later exhibited BAL changes apparently synchronous with small continuum level changes. Despite this apparent conflict, ionizing flux variability of the continuum source is still the best explanation for the BAL changes. The simultaneous changes over a range of outflow velocities observed in several BALQSOs suggest that the mechanism inducing the BAL changes must propagate through the BAL region at (or close to) the speed of light. Also, there is a correlation between those QSOs which show continuum level changes and those which show BAL variations. From the changes observed in various ions (e.g. C IV lambda-1549, N V lambda-1240, Si IV lambda-1397, and Al III lambda-1857) in a few BALQSOs, we deduce that the ionization parameter (the ratio of the densities of ionizing photons and electrons) must be relatively high (U>0.05). We have also estimated a lower limit on the electron density of 104 cm-3 from limits on the response time scale of the BAL changes. From these limits, we deduce that the absorbing gas cannot be further than a few hundred parsecs from the ionizing source. In the few variable QSOs where the continuum slope (alpha, where Lnu proportional to nualpha) was measured, our data indicate that the slope increases as the continuum level increases. Our data also show that the intensities of the permitted broad emission lines often do not appear to follow changes in the observed continuum. This suggests that either this emission arises in a region larger than about one light year, or that the emitting clouds are not ionized by the continuum source seen along our line of sight to the QSO.