The impacts of climate change on precipitation and drought characteristics over Bangladesh were examined by using the daily precipitation outputs from 29 bias-corrected general circulation models (GCMs) under the representative concentration pathway (RCP) 4.5 and 8.5 scenarios. A precipitation-based drought estimator, namely, the Effective Drought Index (EDI), was applied to quantify the characteristics of drought events in terms of the severity and duration. The changes in drought characteristics were assessed for the beginning (2010–2039), middle (2040–2069), and end of this century (2070–2099) relative to the 1976–2005 baseline. The GCMs were limited in regard to forecasting the occurrence of future extreme droughts. Overall, the findings showed that the annual precipitation will increase in the 21st century over Bangladesh; the increasing rate was comparatively higher under the RCP8.5 scenario. The highest increase of rainfall is expected to happen over the drought-prone northern region. The general trends of drought frequency, duration, and intensity are likely to decrease in the 21st century over Bangladesh under both RCP scenarios, except for the maximum drought intensity during the beginning of the century, which is projected to increase over the country. The extreme and medium-term drought events did not show any significant changes in the future under both scenarios except for the medium-term droughts, which decreased by 55% compared to the base period during the 2070s under RCP8.5. However, extreme drought days will likely increase in most of the cropping seasons for the different future periods under both scenarios. The spatial distribution of changes in drought characteristics indicates that the drought-vulnerable areas are expected to shift from the northwestern region to the central and the southern region in the future under both scenarios due to the effects of climate change.