Ramsey theory consists of many deep results that roughly say that any sufficiently large structure contains a well-organized substructure. Early arithmetic theorems of this sort were proved by Hilbert, Schur, and van der Waerden. However, how large is sufficient in these results remains a challenging open problem.  For example, the van der Waerden number w(k;r) is the minimum positive integer N such that every r-coloring of the positive integers up to N contains a monochromatic k-term arithmetic progression. That these numbers exist was proved by van Waerden a century ago, but his proof gives an enormous upper bound on these numbers. Trying to estimate these numbers has remained a major challenge that has led to the development of important methods in analysis, algebra, probability, number theory, geometry, logic, and combinatorics. I will discuss these methods and their implications for van der Waerden numbers and variants, focusing on recent progress.