There is something hypnotic, and more than a little unnerving, about watching from the back seat of a car as the steering wheel moves on its own.

As I was driven through the busy streets of San Francisco by a computer I initially felt like shouting "Look out for that kid" or "Whoah, bike!" But I soon began to realise there was no need. My fully autonomous Uber had already recognised every potential danger and reacted faster than I ever would have. Before I could say "traffic cone!" the wheel jerked left, and then corrected without fuss. As an inveterate back seat driver, I was firmly put in my place. The computer clearly knew best.

It has been a long road to the future but the self-driving taxi is finally here. This week, residents of San Francisco, the home of Uber, found themselves being picked up by cars with a mind of their own.

A small fleet of specially engineered Volvo XC90 SUVs joined the ranks of people-driven Ubers, and will be randomly matched to customers.

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To calm passengers' nerves there will still be a "safety human" in the driver's seat. His coffee cup holder has been replaced with a very large, bright red button, and pressing it will allow him to take over manual control in an emergency.

I was the first British newspaper journalist to try one of the self-driving vehicles, which Uber proposes to operate in an 18sq km area of the city.

My ride began at the understated Uber nerve centre next to a motorway and we took a journey of a few kilometres at random to a pizza restaurant.

After a few minutes on the road, I asked my safety human when he was allowing the car to take over. "It's already the car," he said.

I checked my seat belt and began looking for potential sources of collision. Before long an airport shuttle bus pulled out of a parking space and stopped inexplicably across a lane of traffic. "Er, bus ..." I began to say. But the car had already proceeded to veer smoothly around it without hesitation. I could have sworn it was showing off.

Likewise, a quick quarter-turn of the wheel sent us sailing safely by a double-parked delivery van without braking.

Then, as we approached a right turn, we switched lanes and the blinking sound of an indicator started nice and early. Some people began hurrying across the street. The car stopped and waited for them, more politely than I ever would have. It hasn't been programmed to honk its horn at people who ignore pedestrian crossings.

All this was made possible by an array of seven external cameras and a spinning laser called a lidar on the roof. The system builds a real-time 360-degree view of what is going on around the car.

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The passenger can watch what the car "sees" on a screen in the back. It shows a blurry digital readout in which other cars, bikes and pedestrians are coloured orange. Buildings are blue. It easy to be drawn into watching the screen and forget that this is not a video game.

The screen also has a small "Request Stop" button in the bottom corner in case a passenger begins to feel uncomfortable and wants to get out.

Impressive as it was, there are still numerous issues to overcome, and on more than one occasion my safety human took over. Once, we encountered a stationary car in the road. It was unclear why it had stopped. The Uber halted too and waited patiently, perhaps too patiently, before the human took over and drove around. On another occasion the Uber turned a corner and there were people in the road. It began stopping but the human took over anyway out of an abundance of caution.

The people themselves were somewhat agog as they watched the self-driving car, topped by its helmet of cameras and lasers, go by.

At one point we drove down a busy street alongside another member of the fleet in a kind of driverless parade. The other Uber overtook mine, suggesting individual cars might have differing views of what is an appropriate speed.

The trial is taking place in an area of San Francisco that has been extensively mapped by Uber's cameras. It was chosen for its challenges, which include a preponderance of cyclists, hills, narrow lanes, congested traffic and construction work.

Another trial began three months ago in Pittsburgh but the self-driving cars there were only available to a small group of Uber customers. People ordering one in San Francisco will be able to reject the self-driving option if they don't want it.

The company says it is all part of a learning process. Any time a safety human has to take over the controls, it is regarded as an incident to be studied, and they set about developing computer software to deal with the problem.

The most difficult situations involve human non-verbal communication such as eye contact. An example would be a police officer directing traffic with small hand movements or nods of his head.

Self-driving cars don't learn from encountering a new problem on the roads and have to be programmed how to cope next time.

The impending winter will provide data on how they manage in bad weather conditions such as snow. Engineers already seem confident they can "do rain".

Ultimately, the safety humans will disappear, but not for a few years yet.

One day Uber envisages cities where people don't own cars but just jump in shared self-driving cabs.

Anthony Levandowski, head of Uber's advanced technology group, said: "While it won't happen overnight, self-driving will be an important part of the future of transportation.

"Seven years ago the idea that you could push a button and get a ride was unthinkable."

Uber also believes it will be making a valuable contribution to road safety. Levandowski said that "1.3 million people die a year in car crashes around the globe, and 94 per cent of crashes in the United States are due to human error".

On my way to the company's self-driving research hub, I took a regular Uber to see the contrast. Faisal, my human driver, was a little dubious when I told him I was about to try his computerised counterpart.

"I think the technology is do-able," he said.

"But they've had autopilots in planes for 30 years and you still need a pilot. The human has eyes and can react. How does the machine watch out for crazy drivers?

"And maybe I haven't got as smart a brain but if there's a kid there in the way, I'll go off the side of the street. Will the machine do that?" I had to break it to Faisal that, yes, the machine probably would do that, and more safely.

Still, riding with him reminded me of something that is missing from a self-driving Uber.

It clearly knew more about the roads of San Francisco than any human, but it didn't have a name or a personality.

I give it five stars for being a stunning example of science fiction brought to life. But I also gave Faisal five stars because I enjoyed his company.

Maybe there's room in the future for both of them.