Three to four days covers the historic center, the Vatican and a couple of neighborhoods at an unhurried pace.

Three to four days covers the historic center, the Vatican and a couple of neighborhoods at an unhurried pace.
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The historic center is compact and mostly walkable.
The area immediately around the Eiffel Tower and Sacré-Cœur has persistent souvenir sellers and overpriced cafés; walk a few streets away for better prices and a calmer pace.
Four to five days covers the main museums, a West End show and a couple of neighborhoods.
The Underground (Tube) reaches almost everywhere and is the fastest option; buses are slower but scenic and cheaper.
Restaurants with picture-menus and touts standing outside them right by the major sights are aimed at one-time visitors, not good food.