Look at the things that are dripping. Most likely they are water tanks (cisterns) in the loft, unless they are WC cisterns.
You will probably find they have ball cocks/float valves like this
which I am sure you have seen in WC cisterns.
If the plastic (or if very very old, copper) ball is underwater, or if, when you push it down, you find it is below its natural float point, then the valve needs replacing (you can repair or rewasher them, but a new one only costs a few pounds)
there is a very very slight chance that the adjustment of the float is too high, but this is so unlikely that I wouldn't worry about it if I were you.
The level in the large (cold water tank) should be a couple of inches below the overflow. The level in the small one (central heating boiler feed and expansion, if you have one) should be adjusted to give just a couple of inches depth (so lots of room for expansion).
(If - as Noodlz asks - it is coming from the boiler itself then it may be more serious...)
My girlfriend has just called me to tell me that we have a very slight leak in the airing cupboard (directly below the tanks in the loft) so I presume this could all be linked(?)
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