Solstice: not really. A menstrual cup does not necessarily surround the cervix. Some people have it sit lower in the vagina. It depends on your anatomy, really. It just works for different people in different ways! It collects the menstrual fluids and you dump them out periodically (how often depends on the capacity of the cup and heaviness of your flow) into the toilet (or outside, if you're woodsy like that). As long as the fit is good, there should be very little mess (mostly on removal) and no leaking.
I still can't get mine to sit well. I think the size of the small lunette isn't good for me, but I'm not sure in what way. I've been using Instead Softcups with some success for the last few cycles, along with cloth pads.