Though tensions with North Korea have been stressed the last few years, in reality they are still officially at war with their southern neighbors since the Korean War started in 1950. And while they may be less than hospitable to each other, a "hot" war with actual fighting hasn't occurred since the Armistice in 1953. Still, the tension has mounted and the border is as heavily guarded as ever, making this war one of the longest lasting in recent history.
Another example of a perpetual war today is the United States' infamous "War on Terrorism." Due to the enemy being a faceless, nameless idea (I.e. terrorism), with no physical location, the troops continue to push on into hostile territories indefinitely. The fighting has calmed down, but the American people are still looking for that "Exit Strategy" from this perpetual war.
