Dr Flora Whitmarsh, Science Communications Research Analyst in Climate Change at The Grantham institute, Imperial College London.
We are already getting more high temperature extremes, and probably more heat waves, as a direct result of the warmer mean temperatures. Heavy rainfall is also expected to become more frequent, for reasons that are well-understood.
Warmer temperatures increase the amount of water vapour in the atmosphere
The Earth system tends towards an equilibrium in which about the same number of water molecules are evaporating from the Earth’s surface, predominantly the oceans, into the atmosphere as are condensing from it. If the average temperature of the Earth is increased, the water molecules will have more kinetic energy on average, meaning more of the liquid water molecules evaporate. The system then reaches a new equilibrium with more water vapour in the atmosphere.
The Earth’s atmosphere is made up of several constituents: oxygen, nitrogen, argon, water vapour, and others. Atmospheric pressure refers to the pressure due to all of these components. The saturation vapour pressure is the part of the atmospheric pressure that is due to water vapour, when the system is in equilibrium as described above. This increases with temperature.
The Clausius-Clapeyron relationship is an equation for the saturation vapour pressure, which can be derived from the first and second laws of thermodynamics and has been extensively verified in the laboratory. It says that, at typical temperatures for our atmosphere, every degree Celsius of temperature rise leads to a 6-7% increase in the amount of water vapour in the atmosphere.
What does this mean for average rainfall?
In practice, global annual mean rainfall is only projected to increase by around 2-3% per degree of temperature rise. This is because rain formation is limited by energy constraints. When water vapour condenses to form cloud droplets, energy is released due to the phase transition from gas to liquid, and it takes time for this energy to be radiated off to space. The extra energy in the cloud means less water vapour will condense and the formation of cloud droplets slows down. The IPCC Working Group I Report did not find evidence of any significant change in precipitation averaged over all global land areas over the past century. However, incomplete data meant that confidence was low prior to 1950 and medium thereafter.
It is worth noting that the global distribution of rainfall is also changing, with some areas expected to get wetter and some dryer due to changes in atmospheric circulations, so the 2-3% increase in the mean per degree of temperature rise does not necessarily tell us what will be experienced in any particular region.
What about heavy rain?
However, the intensity of the heaviest rainfall events is only limited by the total amount of water vapour available in the air flowing into the weather system causing the rain. Globally, the heaviest rainfall events are becoming even heavier than before.
The recent Intergovernmental Panel on Climate Change (IPCC) Working Group I report said it was likely  that globally since 1950, the frequency, intensity, and/or amount of heavy precipitation had already increased in more land areas than it had decreased. Furthermore, there had likely been increases in the frequency or intensity of heavy precipitation in Europe specifically. The European increase occurred mostly in winter, and there were decreasing trends in some regions such as northern Italy, Poland and some Mediterranean coastal sites. The report also said that by 2081-2100, it is very likely  that there will be a further increase in heavy precipitation over most of the mid-latitude (temperate) land regions  and over wet tropical regions. An increase is likely over most land areas by 2016–2035.
 In IPCC terminology, likely means 66–100% probability.
 90-100% probability.
 The mid-latitudes lie between the Tropic of Cancer and the Arctic Circle in the north; and between the Tropic of Capricorn and the Antarctic Circle in the south.
Grantham Institute Note on the Changing Water Cycle according to IPCC Working Group I: Changing water cycle - Grantham Note 5 [pdf]
IPCC Report, Working Group I: The Physical Science Basis. 2013.http://www.climatechange2013.org/report/full-report/
Physics of Climate. Jose p. Peixoto and Abraham H. Oort. American Institute of Physics. 1992.
Held, I.M. et al., 2006. Robust responses of the hydrological cycle to global warming. Isaac M. Held et al., 2006. Journal of Climate. Vol 19, pp 5686-5699. http://journals.ametsoc.org/doi/abs/10.1175/JCLI3990.1
Allan, M.R. and Ingram, W.J, 2002. Constraints on future changes in the hydrologic cycle. Nature, vol 419, pp 224-590.