1 Battery Park City - nice promenade, great water views, a lot of trees etc. Nice family place. Far from everything (including subway)
2 Chelsea, Village - old houses,bohemian place.
4 East Village - see Village, is being overtaken by NYU campus recently
5 Financial District - IMHO it's a great place to buy right now. In 10 years this place will become what Tribeca is today
6 Gramercy - have no idea
7 Greenwich Village - see Village
8 Harlem - you don't want to be there
9 Inwood/Washington Heights - you don't want to be there
10 Midtown East - expensive, far from transportation, UN people like to live there
11 Midtown West - I personally don't like that place, it is somewhat nasty
12 Morningside Heights - Columbia University campus. Pretty safe, nothing special.
13 Murray Hill - very expensive
14 Tribeca, Soho - if I'm not mistaken, Tribecca is the most expensive zip code as of right now
15 Tribeca/Financial District see above
16 Union Square - lots of fun, restaurants, stores, cafes etc. Must be a great place to live.
17 Upper Eastside - overpriced. This is there old money like to live. Transportation is a problem. But - close to Central Park and tons of restaurants.
18 Upper Westside - it is nice. You are getting the same access to Central Park and restaurants minus all the hype of the UES.
P.S. Я понимаю что вопрос получился длинноват, но думаю многим будет интересно узнать о Манхэттане подробней.
you can write a book about each of those neighborhoods. So try to be more specific
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