There's no real trick to avoiding crowds at the always-packed museum. The best you can do is try to go in the off-season, early or late in the day, and on a weekday. Your chances of being alone with the Mona Lisa will still be slim to none, but you might be able to actually see that enigmatic smile behind the Plexiglas.
It was Francis I, a great patron of the arts, who built the Louvre Palace on the site of an old fortress and acquired the Mona Lisa; the Louvre was a royal residence for more than two centuries before French revolutionaries turned it into a public museum. Today it displays 35,000 works of art—too many to see in one visit. Wander aimlessly, choose a wing or sign up for a 90-minute guided tour of the highlights at the desk under the pyramid. Closed Tuesdays.