When we sleep, our brains process the events of the day and pick out things to stow away in our long term memory.
Maybe as a side-effect of this processing, they also aid problem solving - after a good night's sleep, you are able to find solutions to the previous day's problems faster (than you would, if you hadn't had the sleep).
As we sleep, our brains clear away toxins that accumulate during our awake time. These toxins may be culprits behind neuro degenerative diseases.
There seems to be a correlation between sleep deprivation and cardiovascular diseases.