Macaulay proved that for every homogeneous ideal, there exists a lex ideal with the same Hilbert function. This theorem is often stated in terms of O-sequences or an inequality involving binomial coefficients. One of the key techniques to proving Macaulay's theorem is a combinatorial inequality that provides a lower bound on the growth of shadows in the infinite grid Nd. This result has had an enormous influence on algebra and combinatorics over the past century. Generalizations of Macaulay's Theorem will be presented.