This was Cyprus' most important pre-Christian city, and it's still the country's best archaeological site. You could easily spend a day here, checking out the fully restored Roman amphitheatre, the gymnasium with marble baths, and the fascinating mosaics.
Salamis' history provides a cross-section of the many civilisations that have contended with each other over thousands of years in the Mediterranean. Subject to the Assyrian, Persian and Ptolemaic empires at various times, its own high period was the 6th century BC. Most of the ruins are Byzantine or Roman, and they're scattered over 8km (5mi). A very pleasant beach fringes the site, so bring your bathers.