Stratospheric water vapour (SWV) is an important component of the Earth's atmosphere as it affects both radiative balance and the chemistry of the atmosphere. Key processes driving changes in SWV include dehydration of air masses transiting the cold-point tropopause (CPT) and methane oxidation. We use a chemistry–climate model to simulate changes in SWV through the 21st century following the four canonical representative concentration pathways (RCPs). Furthermore, we quantify the contribution that methane oxidation makes to SWV following each of the RCPs. Although the methane contribution to SWV maximizes in the upper stratosphere, modelled SWV trends are found to be driven predominantly by warming of the CPT rather than by increasing methane oxidation. SWV changes by −5 to 60 % (depending on the location in the atmosphere and emissions scenario) and increases in the lower stratosphere in all RCPs through the 21st century. Because the lower stratosphere is where water vapour radiative forcing maximizes, SWV's influence on surface climate is also expected to increase through the 21st century.