A 2013 study from the Oxford Martin School estimates that almost half of U.S. jobs could be automated in the next two decades. (PDF)
Of course, whether those jobs would be replaced by new industries is a separate question.
A summary of the study by the MIT Technology Review has this to say about one of the study’s conclusions:
“The authors believe this takeover will happen in two stages. First, computers will start replacing people in especially vulnerable fields like transportation/logistics, production labor, and administrative support. Jobs in services, sales, and construction may also be lost in this first stage. Then, the rate of replacement will slow down due to bottlenecks in harder-to-automate fields such engineering.
This “technological plateau” will be followed by a second wave of computerization, dependent upon the development of good artificial intelligence. This could next put jobs in management, science and engineering, and the arts at risk.”