Some root beers contain caffeine while others don’t. One example is the A & W root beer which has no caffeine content while Barq's root beer contains 18 milligram of caffeine in it. Here's the link of more information: http://www.mayoclinic.com/health/caffeine/AN01211.
Most root beer has no caffeine content. However, there are two brands of root beer identified to contain caffeine which include Bawls Geek Beer and Barq’s. For more information, visit http://www.ask.com/wiki/Root_beer