We have recently been through a whole world of pain in this exact area.
My hubby is fairly handy with DIY stuff.
We got roofers in but they were a disaster and actually made the problem worse.
The real difficulty with a flat roof is figuring out where the leak is coming from.
It often doesn't correspond to where its staining on the inside.
Checking obvious things like flashing etc like the pp said are is a good idea and easy enough to fix.
Using a hose up on the roof with someone supervising the leaking area from the inside, is also a good way of diagnosing the problem.
Any way we bought a roll of roofing felt and watched lots of videos on youtube how to do it and rented a blow torch and gas for a week. It took a while but eventually we fixed it good and proper. Cost about €150 all in.
It is quite dangerous heating roofing felt as you can set fire to it.
You need to practice with the torch a bit on the ground first, before you set fire to the house!!
That's how they started the fire in Guinness's I believe..a blow torch and roofing felt.
A few years ago I fixed a small problem (hubby was away) with some bathroom sealant on the roof...not pretty but lasted a few years