Poets & Writers, is probably best known for its magazine for, well, poets and writers. In fact it describes itself as the nation’s largest nonprofit literary organization serving poets, fiction writers, and creative nonfiction writers. They have two offices, one in New York City and the other in Los Angeles.
Among the many things they do to support writers is to maintain the best database of contests, literary magazines that I’ve come across. Check out their site to find out what they do. And then link to their database Writing Contests, Grants & Awards, to find out where you should be submitting next.