Othello Castle — officially the Citadel of Famagusta — stands at the northern end of the old walled city, controlling the harbour entrance. It was built by the Lusignans in the 14th century and substantially reinforced by the Venetians in the early 16th century, who carved the lion of Saint Mark into the main gate tower — it's still there above the entrance.
The castle is a solid piece of medieval military engineering: round towers, deep vaulted rooms, a Great Hall that was once an impressive interior space and is now a roofless shell. The walls are thick enough to have a road cut through them for artillery. Climbing to the top of the towers gives good views over the harbour, the sea, and the rooftops of the old city.
It is open daily; admission is charged. Allow 45 minutes to an hour to explore properly. The combination of the Venetian harbour-control position, the Shakespeare connection, and the sheer solidity of the construction makes this one of the more rewarding castles in the eastern Mediterranean.