It will begin with advanced cruise control systems, ones that in addition of controlling the accelerator can also control the brakes... those exists already today... and then they will also be able to control steering to keep you in a lane while in the highway.
At this point the issue of liability in case of an accident is pretty much non-existent. It wold be the same as using cruise control. If you crash while using cruise control it is your fault, you are the one behind the wheel... additionally, we would have much more information on the accident given that the car was recording everything in its surroundings as well as copious amounts of data regarding the car's condition, speed, brakes and so on.
Eventually (in the next 5-10 years) we will see how many lives this advanced cruise control systems have saved and we happily will allow total control to machines. (Some estimates indicate that with just 20% of cars on the road with this system we will save 10,000 lives a year in the US alone) By the year 2025, the transition to completely self driving cars will be welcomed by all.
And those are mostly young healthy lives too. This is different than saving lives of people with cancer or Alzheimer. These are lives that will keep on ticking for many years to come.
And the amount of money saved? in the billions. Insurance companies will go out of business (not really, they'll figure out a way to keep their hands in our pockets) But the money saved to society as a whole will be huge.
Twenty years from now, by 2035, we... I hesitate to talk about 20 years from now. Most people don't think that far and they look at me like I'm crazy when I tell them what I think the world will look like in 20 years.
But let's do this. Let's look at 20 years in the past. Google did not exist. Neither did Wikipedia. We actually had bets with friends about what was the name of the actor in such and such a movie. And those bets would take days to get resolved. Nowadays you take your Android phone, click on the microphone next to the search box and say "what's the name of the actor in the Ghostbusters?" and the entire cast of the movie is displayed in our phone including photos and links to clips and perhaps even a link to the actual movie.
Now take that look back and multiply it forward...
But Wait!
It's not linear, arguably it's not fully exponential perhaps... but it's certainly not linear. If it was exponential we would have to look at how technology advanced in the last 200 years to get an idea of how much it will advance in the next 20. Then your mind would be blown. So to imagine what would happen in 20 years we need to look farther back than 20, but perhaps not as much as 200. Maybe 50 years, maybe 40?
I may be off by a couple of years -although I think my estimate is a bit too conservative- but self driving cars will be here soon and we will be happy about it.