If you are looking for a new apartment in Manhattan, chances are that you are going to be looking for homes in neighborhoods like the Upper West Side or Greenwich Village. After all, these are the neighborhoods that are enjoying a lot of buzz, thanks to both the quality of the homes in the area, as well as the livability of the area itself. Having said that, the apartments here won’t come cheap, since rents, after all, are so closely correlated with the reputation of the neighborhood. But there are exceptions to the rule—there are some Manhattan neighborhoods that are currently on the upswing, and the rents there haven’t shot through the roof—yet.