Borel ideals are monomial ideals closed under the operation of replacing a variable with one of a smaller index. Apart from an algebraic interest, Borel ideals satisfy many nice enumerative properties. We'll talk about a generalization of the Borel property which includes all monomial ideals. We'll also show links between Borel ideals and Catalan numbers, certain planar point configurations, and binary trees. (This is joint work with two other Cornellians, Chris Francisco and Jeff Mermin.)