Piazza Umberto I: Also known as the Piazzetta, this pretty square is Capri Town's hub and is lined with chic cafes. The 18tth-century Baroque cathedral, the picturesque clock tower and the town hall are all worth a look.
The Blu Grotto: The walls of this limestone cavern, Capri's most famous geological phenomenon, turn a deep and sparkling blue by the reflections of the sea which fills it. To add to the effect, the local boat operators who take tourists in there on small motorboats demonstrate the cave's echoing acoustics by giving hearty renditions of 'O Sole Mio.' Great fun, but beware of banging your head on low rock faces, and be prepared to have your trip canceled if the seas are too rough.
Villa San Michele in Anacapri: This is the spectacular former home of Swedish writer Axel Munthe. Raised from the ruins of a villa built by Emperor Tiberius, it has glorious gardens and stunning sea views, and contains an impressive collection of Roman artifacts and 17tth-century furniture, as well as a statue of Hermes given to Munthe by the city of Naples in gratitude for his work in the city during the cholera epidemic of 1884. (+39 081 8371401; open daily 9 a.m. to 6 p.m. May to September, 9 a.m. to 5 p.m. April to October, 9 a.m. to 4:30 p.m. March and9 a.m. to 3:30 p.m. November to February)
Giardini di Agosto: You'll find these lovely gardens by walking from the main square of Capri Town down Via Vittorio Emanuele. The park offers wonderful sea views and overlooks the Faraglioni (rocks which form three pinnacles as they rise from the sea).
Hit the Shops: The grandest boutiques in Capri Town are to be found on Via Vittoria Emanuelle, Via Camarelle and Via Tragara. If you're really determined to splash out, or just want to daydream while viewing some serious rocks, check out La Perla Gioielli on Piazza Umberto I -- it's Capri's most prestigious jewelery store (and, we suspect, another of Liz Taylor's favorite places).
Get High: Enjoy dizzying sea views from Anacapri and Mount Solaro. You can go on foot (if you are very fit) or take the Monte Solaro chairlift (free to children under 8). The chairlift entrance is on Piazza Della Vittoria in Anacapri, and the ride takes about 12 minutes.
Hike: Go on a hike and enjoy spectacular views every time you stop to catch your breath. Maps featuring suggested walking routes can be picked up for about two euro from tourist booths at the pier or on Piazza Umberto I.