Yes I would imagine its part of my monthly allowance but then so are calls and they will probably cost more from abroad I have International roaming set up already
Turn off the mobile network as soon as you get on the plane and rely on wifi hotspots. It will cost you a fortune if you don't, especially if you have apps that automatically update. Some games, weather and news apps do that, and of course email.
I contacted my phone provider and insisted they put a £50 limit on my monthly bill. Everyone should insist their limit is adhered to