Parco Archeologico della Neapolis

Built 60-40 BC under Augustus or Julius Caesar, this is mostly carved out of the hillside, with no superstructure surviving. So effectively it's a grassy pit 140 m long by 119 m wide, and you'll have to use your imagination to see the tiered seating and walkways. The pit in the centre held machinery for stage effects.

This massive piece of rock, 199 m long by 23 m wide, was the foundation block of an altar, the largest known from the ancient world. It was built by King Hiero II (who had Archimedes in his employ) some time before 200 BC. Nothing else remains of the temple here; classicists believe it was probably built to the glory of the deity Zeus, and definitely built to glorify Hiero himself.

From the entrance, follow the driveway to descend into a great bosky hollow. These are the ancient stone quarries, worked on from the sixth century BC. Two grottos or galleries burrow into the rockface. Largest is Orecchio di Dionisio or "Ear of Dionysius". Some grotto! - its dimensions are worthy of the gothic nave of a city cathedral. The name arises from its acoustics: the legend says that the tyrant Dionysios could stand outside and listen to whispered conversations of those imprisoned therein. Difficult to believe this, once you've savoured the distorted echoing babble of tourists.
The second and smaller is Grotto dei Cordari ie the cave of cordwainers or ropemakers. Dark, not particularly straight, far from dry, so it's not obvious why they would work here.
At the very north end of the quarries are Necropoli Grotticelle - which supposedly contains Archimedes' tomb, but the site of this is unknown. It's probably easier to view from the street after leaving the park.

Built circa 470 BC but with reconstruction in the 3rd century BC and Roman period. Further work continues today, so you can look down on it but can't enter: they're rebuilding the stage, and a tier of wooden bleachers. Greek drama is performed here May - July, typically at sunset. These performances are in Italian, without amplification as the natural acoustics are so good. It is the largest remaining Greek theatre on Sicily, but there isn't much left other than the seats.

An underground necropolis dating back to the 4th century AD.

This church is a huge concrete cone built from 1966 but only completed in 1994, inspired by a mass-produced Madonna statue that is said to have cried real tears in 1953. The interior is a great circular expanse, with six of the 20-some side spaces dedicated as chapels, with the tower above like some colossal machine part. The lower floor or crypt is an equally large circular space.