By scanning the brains of people who were asleep, the researchers have found that the same areas which are involved in learning new tasks while a person is awake are still busily processing information while they are asleep.
The finding suggests that sleep allows our brains to store new information in the memory for future use.
People in developed countries are spending less and less time sleeping - down from an average of nine hours a night in 1910 to about seven hours today.
But without enough sleep, wounds and injuries take longer to heal, and it is difficult to concentrate or learn new tasks.
The scientists, led by Pierre Maquet of the University of Liege in Belgium, used a device called a positron emission tomography (Pet) scanner to monitor the brain's electrical activity.
The Pet scanner works by measuring the rate at which areas of the brain use up glucose and oxygen - high levels indicate activity.
New task
They gave a group of 18 volunteers a new task to learn, and scanned them while they were awake. Then they allowed the volunteers to fall asleep, and scanned their brains again.
They found that the pattern of activity was very similar during the brief but active stage of sleep known as rapid-eye movement (REM) sleep.
When those volunteers awoke, they were able to perform the tasks better than before they had gone to sleep. The researchers said the volunteers might have been practising the test in a REM dream and storing what they learned.
However, they could not identify the precise cellular mechanisms involved. Neither do they think that memories are only stored during REM sleep.
Previous research has found similar results in rats, but the human brain is much more complex.
David Silbersweig, co-director of the functional neuroimaging research laboratory at the New York-Cornell Medical Center, said: "It is wonderful to see such results demonstrated in humans for the first time."