In reply to mlt:
I went ski mountaineering in Georgia last spring, to Kasbek and Mestia. We weren't asked for any permits anywhere. We stayed in the betlemi hut, a huge disused meteo station, for the skiing around and on Kasbek and there is a charge for this, ask at the Geoland map shop in Tblisi.
In terms of safety we felt completely fine save for one incident which was completely our fault. We had a celebratory night out in Tblisi after successfully skiing Kasbek and decided to drink cha cha, the local spirit. This led, through a rambling set of circumstances, to two us being robbed at knifepoint and having phones, wallets and stupidly, my passport being taken. I thought this meant that the rest of my trip would be spent in an embassy trying to get home, but the police arrived soon after the robbery and, through another set of even more rambling circumstances, found the thieves and all of our stuff! The police were truly excellent.