In this pre-registered study, we analyze the effects of need-based financial aid grant offers on the educational outcomes of low-income college students based on a large-scale randomized experiment (n=48,804). We find evidence that the grant offers increase two-year persistence by 1.7 percentage points among four-year college students. The estimated effect on six-year bachelor's degree completion is of similar size--1.5 percentage points--but is not statistically significant. Among two-year students, we find positive--but not statistically significant--effects on persistence and bachelor's degree completion (1.2 and 0.5 percentage points, respectively). We find little evidence that effects vary by cohort, race, gender or the prior receipt of food stamps. However, further exploratory results do suggest that the offers reduce associate's degree completion rates for two-year community college students by around 3 percentage points, with no statistically significant evidence of effects on technical college students. We also estimate that the effects of actually receiving grant money are very similar, though slightly greater than the effects of merely receiving a grant offer. Overall, our results show only very small effects of the need-based grant offers on college students' trajectories towards degree completion. [Additional funding from the Fund for Wisconsin Scholars (FFWS).]