Amorgos is a fashionable yet decidedly low-key island in the eastern Cyclades. Its pristine beaches are wonderfully secluded and many of them can only be reached by foot or by boat. The island boasts some world-class dive sites, as well as numerous picturesque trails along its rugged coastline. Make your way inland and you’ll find the quintessentially Cycladic town of Hora. Lose yourself in its narrow streets, charming squares and bougainvillea-draped whitewashed buildings. Back on the coast, visit the dramatic 10th century Monastery of Hozoviotissa, which clings to a cliff some 300 meters above the Aegean. In the evening, chill out in a traditional taverna in the main port of Katapola, or head to the secondary harbor at Aigiali, which draws a livelier crowd.