There are two main approaches: taxis and personal vehicles. There are many companies that have gotten as far as testing with a safety driver and/or with "trusted tester" riders, but as far as I can tell only two [EDIT: three? four?] companies run commercial ride services open to the public without anyone in the driver's seat:

• Waymo (Google-affiliated) has been operating fully driverless vehicles in Phoenix AZ as a commercial ride service since 2020. They used to have a waitlist, but at this point anyone can download the app and try it. They've just expanded to SF, and have a waitlist for LA and Austin. As of 2023-08 they claim to be serving 10k weekly riders.

• Cruise (GM-affiliated) has been operating fully driverless vehicles in SF as a commercial ride service since 2022. They also now seem to cover Austin and Phoenix, and as of 2023-08-02 also claim 10k weekly riders.

• [Added 2023-08-23] It's hard to tell because the best sources are in Chinese, but it looks like Apollo (Baidu) meets this as well. As of the end of 2022: "In addition to Wuhan, Baidu now provides commercialized autonomous ride-hailing service with no driver or safety operator in Chongqing as well. Beijing will soon be added to the list."

• [Added 2023-08-23] There's even less English-internet information on this one, but another Chinese company, pony.ai may also be running a fully driverless commercial ride hailing service. "Pony.ai said it has launched fully driverless autonomous ride-hailing services in Beijing and Guangzhou." (2023-03-17)

For personal vehicles, the most automation you can currently get is Level 3. These are systems where, when engaged, the person in the driver's seat can safely and legally read a book or otherwise not pay attention. If the system runs into a situation it can't handle, it alerts the driver, and if the driver doesn't take control it automatically stops the car. One option here, and maybe the only commercially available one, is Mercedes' Drive Pilot, which they launched in Germany in 2022. It only operates on highways under 40mph (essentially, stop-and-go traffic) but Mercedes takes on the legal liability when it's engaged. They claim their 2024 (as in, starting late this year) US models of their S-class and EQS sedans will have DrivePilot as an option and are approved in CA and NV. There may be other L3 systems in other countries—I'm having a lot of trouble telling what exactly launched with what models and whether it qualifies as L3.

Overall, the current state is beyond "it's just a demo", but it's also still heavily limited by location and current conditions.