The Royal Society warns that the risk of heatwaves to an ageing population will rise about ten-fold by 2090 if greenhouse gases continue to rise.
They estimate the risk to individuals from floods will rise more than four-fold and the drought risk will treble.
The report???s lead author Prof Georgina Mace said: ???This  problem is not just about to come??? it???s here already."
She told BBC News: "We have to get the mindset that with climate change and population increase we are living in an ever-changing world ??? and we need much better planning if we hope to cope."
The report says governments have not grasped the risk of booming populations in coastal cities as sea level rises and extreme events become more severe.
???People are increasingly living in the wrong places, and it's likely that extreme events will be more common," Prof Mace says.
???For most hazards, population increase contributes at least as much as climate change - sometimes more. We are making ourselves more vulnerable whilst making the climate more extreme.
???It is impossible for us to avoid the worst and most unexpected events. But it is not impossible to be prepared for an ever-changing world. We must organise ourselves right away."
The report???s team said the UK was comparatively resilient to extreme events ??? but still vulnerable because of the high density of people living in areas at risk.
The report advises all levels of society to prepare ??? from strategic planning at an international and national level to local schemes by citizens to tackle floods or heatwaves.
Its scenarios are based on the assumption that the world stays on the current trajectory of emissions, which the authors assume will increase temperature by 2.6-4.8C around 2090. It assumes a population of nine billion.
They say they have built upon earlier work by calculating the effects of climate change coupled with population trends. They warn that the effects of extremes will be exacerbated by the increase in elderly people, who are least able to cope with hot weather.
Urbanisation will make the issue worse by creating ???heat islands??? where roads and buildings absorb heat from the sun. As well as building homes insulated against the cold, we must also ensure they can be properly ventilated in the summer, the report says.
The authors say cutting greenhouse gas emissions is essential. But they argue that governments will also need to adapt to future climatic shifts driven by climate change.
They suggest threats could be tackled through a dual approach. The simplest and cheapest way of tempering heatwaves, they say, is to maintain existing green space. Other low-cost options are planting new trees, encouraging green roofs, or painting roofs white to reflect the sun.
The authors say air conditioners are the most effective way of keeping cool ??? but they are costly, they dump heat into city streets and their use exacerbates climate change.
Flooding is another priority area, the report says. It finds that large-scale engineering solutions like sea walls offer the most effective protection to coastal flooding - but they are expensive, and when they fail the results can be disastrous.
The ideal solution, the authors think, may be a combination of ???hard??? engineering solutions like dykes matched with ???soft??? solutions like protecting wetlands to hold water and allow it to seep into the ground.
A scheme at Pickering in Yorkshire previously featured by BBC News is held as an example. The report concludes more research is needed to measure the effectiveness of these ecosystem solutions.
It insists that governments should carefully prioritise their spending. They should protect major infrastructure like electricity generation because of its knock-on effect on the broader economy. They should expect some lower-priority defences to fail from time to time, then work to minimise the consequences of that failure .
The authors identify excess heat as another potential threat to economies and agriculture if temperatures climb too high for outdoor workers.
They examine projected rises in the ???wet bulb??? index used by the US Army and others to measure the temperature felt when the skin is wet and exposed to moving air.
Some areas may experience many weeks when outdoor activity is heavily restricted, they fear ??? although the trend of agricultural labour loss may be offset through the century as more and more people move to cities.
It puts a figure on those at greatest overall risk: populations in poor countries make up only 11% of those exposed to hazards but account for 53% of the disaster deaths.
Some economists argue this shows that poor nations should increase their economies by burning cheap fossil fuels because that will allow them to spend more later on disaster protection.
The authors also call for reform of the financial system to take into account the exposure of assets to extreme events.
They say: ???Unless risks are accurately evaluated and reported, companies will have limited incentives to reduce them. And valuations and investment decisions will continue to be poorly informed.???
One author, Rowan Douglas, from the Willis Research Network, said he suspected this might be the most significant contribution of the report.
The authors want organisations to report their maximum probable losses due to extreme events, based on a 1% chance of the event on any given year.
???The 1% stress test is not as extreme as it might sound ??? it implies a 10% chance of an organization being affected once a decade,??? they say.
They say decisions made over the next few decades as the world builds vast urban areas will be key to the resilience of people by the end of the century.
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Climate change and population growth will hugely increase the risk to people from extreme weather, a report says.