The Seine is the equator of Paris, the river that divides Paris into two halves, the left bank and the right bank. The left bank used to be the more fashionable half. It is where Hemingway and Fitzgerald spent their time, as well as the location of the famous bookstore Shakespeare and Co.
The Right bank is becoming as fashionable as the left bank. The Marais, the Bastille, Belleville, and many other neighborhoods are located in the Right bank. So by no means limit yourself to the left bank. You'll find yourself traveling back and forth within a day.
The Seine and The City of Light
As you travel to and fro between the left and right bank, you'll cross the Seine every time. The Seine is almost like a monument. It's beautiful, romantic, and crossing the bridge across the Seine is one of the better experiences in Paris. You can imagine why Paris is called the City of Light when you cross the Seine--you can get a very good view of the city from a bridge over the Seine.
If you're interested in going into the Seine itself there are many boat tours that can show you the sights from the water. The Seine is like Paris's main artery, so there isn't much you won't be able to see. It can even take you clear out of the city.
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