Doha is the home base of Qatar Airways. The government has decided to introduce some sights they regard worth seeing, including a rebuilt Souq and a few museums. From the airport, you can take a taxi into the city center, which costs 50 Qatari Rial (approx. 12,50 euro). Qatar Airways has prolonged its free stop-over program, by which you can get a 4 or 5 star hotel for a night for free (excluding breakfast).
Museum of Islamic Art
As we had a 15 hour stopover during daytime, we decided to get into the city center and visit the Museum of Islamic Art. That museum is well worth the visit, exhibiting a beautiful, though (chronologically and geographically) diverse collection of artefacts, in an impressive building. Entrance is for free, and the museum has a cafe with reasonably priced food and drinks, and a shop with less reasonably priced stuff.
Honestly, do you want to see this giant pearl?
But otherwise, to be honest, I wouldn't bother again to visit the city. Basically, it has the feel of a suburb. It's usually insanely hot (we once got offered a terrace heater when we sat outside at nights and it was 'only' 24 degrees centigrade) and it hasn't got a 'soul' (at least we haven't discovered it). Qatari people are doing their shopping, followed by their nannies and porters as if they were their shadows. But if you fancy see a giant pearl, well, it maybe something for you...