Before Lincoln Center changed the cultural geography of New York, this was the home of the New York City Ballet (originally known as the Ballet Society). City Center’s lavish decor is golden, as are the companies that pass through. You can count on superb performances all year long, including Alvin Ailey American Dance Theater, the Paul Taylor Dance Company and the invaluable Encores! musical-theater series. In September, the Fall for Dance Festival features performances by an assortment of companies.