In addition, the risk of exposure to flooding due to extreme rainfall could more than quadruple, according to the findings.
Scientists calculated the impact of climate change and population changes on the chances of people being affected by floods, droughts and heatwaves around the world.
In the UK and other parts of Europe, a dense and increasingly ageing population meant that heatwaves were the most serious threat.
An estimated 2,000 British deaths were attributed to the warmest summer for 500 years in 2003.
Last year, up to 760 people reportedly died in England alone during a July heatwave.
The new research suggests that such events could become much more common in years to come.
Scientists adopted a “worst case” scenario by assuming an increase in average temperatures around the world of 2.6 – 4.8C by 2100.
But global warming is on course to reach this level unless governments agree to a meaningful strategy for cutting greenhouse gas emissions at critical talks next year.
The researchers defined a heatwave as a run of five days during which night-time temperatures are at least 5C above the norm.
Professor Peter Cox, from the University of Exeter – one of the authors of the Royal Society report, said: “We measure exposure to individuals. That goes up because of more extreme events and because the size of the vulnerable population increases ..
“Climate change increases the risk to people by a factor of two or three and population change multiplies that by at least 1.5 and up to four times in the case of heatwaves.”
In the UK, the proportion of people aged 65 and over is expected to increase significantly – making more of the population vulnerable to higher temperatures.
The report also found a dramatically increased risk of exposure to flooding in the UK and other parts of western Europe, while the threat of drought hung over the Mediterranean.
Climate change is expected to make wetter parts of the world wetter and drier parts drier.
The report issued an urgent call to both governments and private companies to do more to address extreme weather hazards.
Engineering solutions such as dams, sea walls and levees were often costly and also at risk of failing “cataclysmically”, it was claimed.
The experts recommended that engineering be combined with natural ecosystem-based approaches such as re-establishing flood plains, protecting coastlines with mangrove forests, and planting vegetation.
Dr Nancy Grimm, from Arizona State University in the US – a member of the working group that wrote the report, said: “We need to make sure that large-scale engineering isn’t making us too complacent. In the developed world we have been heavily reliant on some key large-scale pieces engineering projects, which have been pushed to their limits during recent events.”
The scientists also warned that unless companies improved the way they handle weather risks their credit ratings could suffer.
Co-author Rowan Douglas, chairman of the Willis Research Network – which advises public and private institutions on risk, said it was important that city planners also factor in the increased likelihood of extreme weather events.
“At a macro level, we will re-build most of the world’s cities in the next 30 years, literally,” he said. “We have a choice whether to build them to be vulnerable or resilient.”
Professor Georgina Mace, from University College London – who chaired the Royal Society working group, said: “Resilience means people are able to do more than just cope with disastrous events. They’re not just absorbing the impact; they’re able to continue living their lives and to prepare for future such events.”
The report did not look at wind damage, which poses the greatest potential risk to property in the UK.