NuTonomy looks to have beaten Uber to the punch. While still very small scale, NuTonomy is actively offering rides (currently for free, like Uber plans to do in Pittsburg) in Singapore.
NuTonomy is a small private company with staff originally from MIT. The cars will currently have a human driver aboard, but the cars will run autonomously. The nuTonomy CEO says this pilot project is to guage how the public responds to self-driving cars.
I am excited to see how rapidly this technology is evolving and being implemented.