Over at the New York Personal Injury Lawyer Blog, Eric Turkewitz has posted a very interesting short comment called Uber Cars are Uber Dangerous. You should read the full comment but the bottom line is that it is possible to argue that the way Uber works, it encourages its drivers to check and reply to text messages while driving. If this is the case, it can then be argued that Uber is acting in a way that creates unreasonable risks of harm to others, and therefore, that it should be liable for injuries that result. This would be particularly important in jurisdictions that have adopted the view that someone who sends a text may be liable if he or she knows that the recipient is driving at the time and is likely to be distracted by the text. There are reported cases on this in at least two states.
It should also be noted that this theory of liability is not dependent on finding that Uber drivers are employees as it would be if the argument is based on vicarious liability (an issue that is currently being litigated in several cases in different jurisdictions).