If you position the mooncup correctly there shouldn't be any leaks because you kind of put it in folded and it springs open inside you and then sucks up against the cervix forming a vacuum seal. So, even if it gets pretty full or you lie down or whatever, if you've formed that seal, it shouldn't leak. Some people I know who have experienced leaks, it has been because that seal hasn't formed, they've just placed the mooncup like a cap. It does also work like that but will leak. To take the mooncup out, you then pinch the bottom to release the seal gently and remove.
Again, this is probably TMI for some, but it's a hard topic to talk about without the TMI thing! There's loads of advice and FAQs on it at http://www.mooncup.co.uk/