When Franklin D. Roosevelt declared war on Japan after the 1941 Pearl Harbor attacks, it signified the last time the U.S. officially declared war.
Korea, Vietnam, the Persian Gulf, Afghanistan, Iraq: technically, those were not wars. Those conflicts, and other in between, are considered "Extended Military Engagements." President Obama too has been selective about the way he uses the word "war" in the build-up to today's situation with the Islamic State in Iraq and Greater Syria (ISIS).
Call it whatever you want: "targeted action," "a systematic campaign," or a "sustained counter-terrorism strategy" - but don't call it war.