Currently (May 11, 2014) the US is not involved in any declared wars, and hasn't been since August, 1945.
The only active conflict is the ongoing "war on terror", in which American soldiers and airmen are fighting Islamic terrorists in Yemen, Afghanistan and Iraq.
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Over the past 200 years, there have been many military conflicts and wars that the US did not participate in.Crimean WarRusso-Japanese WarFranko-Prussian War
If you mean the northern states, they fought in the Civil War. Other than that one time, the US has always been united. Any wars which involved the north, involved the entire US. The US has, of course, fought in two world wars and many other wars.
The two major players in the Cold War were the US and the Soviet Union. These two nations did not have battles or real wars during the Cold War years. Other conflicts among smaller communist nations, and a large one, China, were related to the Cold War. Wars in Korea and Vietnam can be considered wars with roots in the Cold War.
Iraq
Yes: All centuries have had their share of conflicts in various parts of the world. The 1800s are notable for Napoleon, the Crimea, the US Civil War, the war of 1870/71 Prussia vs France, the Zulu wars in Africa, the Boer War.