Future intensification of Amazon drought resulting from climate change may cause increased fire activity, tree mortality, and emissions of carbon to the atmosphere across large areas of Amazonia. To provide a basis for addressing these issues, we examine properties of recent and future meteorological droughts in the Amazon in 35 climate models participating in the Coupled Model Intercomparison Project Phase 5 (CMIP5). We find that the CMIP5 climate models, as a group, simulate important properties of historical meteorological droughts in the Amazon. In addition, this group of models reproduces observed relationships between Amazon precipitation and regional sea surface temperature anomalies in the tropical Pacific and the North Atlantic oceans. Assuming the Representative Concentration Pathway 8.5 scenario for future drivers of climate change, the models project increases in the frequency and geographic extent of meteorological drought in the eastern Amazon, and the opposite in the West. For the region as a whole, the CMIP5 models suggest that the area affected by mild and severe meteorological drought will nearly double and triple, respectively, by 2100. Extremes of wetness are also projected to increase after 2040. Specifically, the frequency of periods of unusual wetness and the area affected by unusual wetness are projected to increase after 2040 in the Amazon as a whole, including in locations where annual mean precipitation is projected to decrease. Our analyses suggest that continued emissions of greenhouse gases will increase the likelihood of extreme events that have been shown to alter and degrade Amazonian forests.