The next one will be automated "city cars" built by Google, that will pickup and drop off people at work and take them shopping and whatnot.
Let's not get ahead of ourselves, such a car has yet to be demonstrated. Google's demo vehicles are incapable of taking riders anywhere apart from a set track of stops, like a Disneyworld people-mover ride.
There's still probably a need in some cities for street-hail livery, which is what classic yellow cabs are -- in NY you can wait 5-10 minutes for the Uber or hail a cab in 30 seconds, and frankly the cabbie will be less of a pain -- my experience with Uber drivers in Manhattan has been a pretty mixed bag. As long as humans are doing the driving it might still be advisable for the drivers to get background checks and have commercial licensing and insurance, such things are prudent and won't kill the magic free market pixies that flutter about e-hailed car services.
As I understand it, city governments have a few simple problems with Uber-
1) Ubers can avoid poor neighborhoods at will, and there's really nothing the city can do about it. I live in LA, and if you live in, say, Watts, you must call a cab if you want a car, no Uber will find you there, because it's "the ghetto" and there's never an Uber within 20 minutes. Taxis can be and are required to pick up from all parts of the city, and their statistics are closely monitored by regulators to make sure they do.
2) Uber's trip pricing structure is very free-markety but it conflicts with most city's basic taxi regs, wherein a trip's price is a fixed formula of distance and time, no special charge for time of day or pickup/destination location. Uber can't provide this, because they use rate premiums to recruit drivers. Again the system is completely open to various kinds of discrimination, and the pricing process is completely private and not open to any sort of public accountability or scrutiny -- even they drivers, who are nominally the service providers ("Uber is not a transportation company"), can't control it.
3) These of course lead to the more philosophical dispute, namely, Uber handles the hailing, transaction processing, driver and rider ratings, and branding of the interaction, but whenever there's any sort of trouble, Uber can vehemently claim they have nothing to do with the driver or the ride, that it's none of their business, and governments and harmed parties must direct all their laws and lawsuits at little sole proprietors. This is a little too clever by half for some people and while following the letter of the law tends to skirt the equities a little too close.
All of this is totally fine as long as e-hail livery is a "premium" service, but some cities rely on taxis as a critical part of the transport infrastructure, and that's when price disparities and availability blackouts start to be problematic, politically.