The University's Pleasance sports centre (often known as the CSE) has bouldering and top-roping facilities open to members of the gym, their guests, University students and staff. You'll nearly always meet yummicks there since it's so close to the main university area and just down the street from GBH. A day pass with gym entry is currently £5, and EUMC members can purchase a discounted pass for a year's climbing.
Further pricing information, times of opening and contact details can be found on their website.
Alien One/Rock (A1)
Housed in a former church building, Alien Rock was Scotland's first dedicated indoor climbing centre. It offers an exciting and interesting range of indoor climbing (top roping, sport, and bouldering!), with roughly 200 routes at a time which are cleaned and changed every four months.
Alien Bloc (A2)
Alien Bloc is a bouldering facility for Alien Rock with 100s of different problems. There are over 500 sqm of varied angled walls from slab through to an outrageous 7 meter long roof. There are hundreds of different climbing "problems", more than enough to keep both novice and experienced climbers entertained for many visits.
Both A1 and A Bloc can be easily accessed by bike or by bus. For more info on Alien visit their website.
Eden Rock is Edinburgh newest bouldering gym, situated in Carlisle, they boast one of the best wooden circuits in the world, for advanced climbers. In addition to the main room, there's a comp wall, circuit board, a dedicated training room and an area for beginners.
Ratho is enormous -- tons of routes up to 30m high, with top-roping, leading, and bouldering all available. It's popular for endurance training, especially in midwinter -- but bring your belay jackets as it can get really cold. Unfortunately, it isn't well served by public transport (two buses and a walk from central), but it's only a 40-minute cycle along the canal.
For more info visit their website.