The Istanbul Archeological Museums hold outstanding collections of art from the ancient Near East, Greece and Rome.
When archeology became a science in the 19th century, most of the Near East was ruled by the Ottoman sultan, who got all the best stuff. These were his private collections, so they're pretty amazing.
The oldest stuff, in the Museum of the Ancient Near East, (Eski Sark Eserleri Muzesi) includes one-of-a-kind treasures like the tiled reliefs of lions and kings from the fabled gates of Ninevah, and the oldest written peace treaty ever discovered (cuneiform tablets which preserved the words of the Hittite and Egyptian kings). (The Egyptian hieroglyphic version of the treaty was carved into stone at the Great Temple of Amon at Karnak, Luxor, Egypt.)
The Archeological Museum (Arkeoloji Müzesi) is devoted to later stuff; ie, after about 500 BC, It includes many of the finest Greek and Roman marble statues discovered in Anatolia.
The Tiled Pavilion (Çinili KoSk) is among the oldest Turkish secular buildings in Istanbul, an elegant residence built by Sultan Mehmet the Conqueror. It's been beautifully restored and is now an outstanding Museum of Turkish Faïence.