This Sultans' palace was built by the Sultan Mohammad II in 1462 and finished in 1478. It was the official residence of the Ottoman monarchs until 1855.
It's a must-see for its gardens, rooms which seems to be in a competition for most opulent decoration, amazing tile-work, impressive views, historical relics and gold, and the seemingly endless amount of small details that make it a truly unique place. I'd suggest taking your time to enjoy ever inch of it!
I'd also suggest getting there first thing in the morning (it gets crowded quickly) and keep in mind that you need two separate tickets: one for the palace and one for the harem.