What city in United States is not located within a state?

The capital of the United States, is Washington DC, is not located within a state. The "D.C." part of it's name stands for "District of Colombia". Our founding fathers did not want the national capital to be inside of any state for conflict of interest.

The flag for Washington D.C. is actually taken from President George Washington's family flag. Which was a white background having two red horizontal lines going across it and three red stars above it.