Obama: Zero. Zip. Zilch.
Obama has never served in any branch of the military, although he does sit on the U.S. Senate Committee on Veterans Affairs. It bears mentioning that he’s too young to have faced the draft during the Vietnam War.
Romney: Religious Exemption
When Romney was finishing up his freshman year at Stanford in 1966, America was about halfway through the Vietnam War and war fatigue was stirring at home and abroad. Romney was restless himself, but for very different reasons. Feeling adrift, he followed his father and grandfather's lead and decided to dedicate three years to Mormon missionary work. His assignment: Le Havre, on France's Normandy coast. Thanks to a special agreement between the LDS Church and the Special Service, Romney was categorized as a "minister of religion" and exempted from the draft. Upon returning to the States in 1969, Romney was granted a student deferment and, ultimately, when the draft lottery was introduced in December of that year, received a high enough number — 300 — that his chances of being called up for duty were practically zilch. That said, he claims to have wanted oh-so-badly to join the fight. In an interview with the Times in 2007, he said: "There were surely times on my mission when I was having a particularly difficult time accomplishing very little when I would have longed for the chance to be serving in the military, but that was not to be."