In reply to ClimberEd:
Hi I've lived in Doha now for 4 months and yes there are both!
There is a climbing community on Facebook called Doha climbers this will tell you the most.
There is only 1 artificial climbing wall here, in the Apsire Zone, but it seems they won't staff or open it, so people have to resort to jumping the fence.
However, there is plenty of outdoor climbing. Some is inland in sink holes at the end of the Salwa Road, the rest is on the coast. It seems there are single pitch top rope climbs, but you will need a 4wd to get there.
You can run along the Corniche which is really nice and is 8km each way. There is also a running and training circuit in the Aspire Zone Park (near Villagio). This is a pleasant grased park (with artificial bird noises) and is full of families, but plenty of peopel use it to workout.
The weather her right now is very nice so it will be perfect for running.
I saw rain for the first time today since moving here!
Dean