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An absolutely must is
Sea Satin, right below the famous Mykonos windmills. It's a very popular restaurant with Greek people where you can taste excellent fish and meat, perfectly cooked on the grill and other Greek specialities 'till late at night (well after midnight). Guests often dance among the tables and sing Greek popular songs together with the owners!
Another place is
Nobu Matsuhisa, inside the Belvedere hotel, if you like Japanese-Peruvian style cuisine with perfect tables put all around a well-lit pool. Booking is essential. Very high prices, but it's Nobu!!!
I want to mention also
Chez Katrine's, in the old town, where you can taste both Greek and French dishes, in an elegant space or right in the narrow street, maybe a but pricier compared to others in town, but it worth going.
M:Eating , in mykonos town near the Alpha Bank. Greek and fusion cuisine at reasonable prices as the food the serve is delicious. Intimate garden on the side or a street side balcony for people watching