  We consider the maximum of the discrete two dimensional Gaussian free field
(GFF) in a box, and prove that its maximum, centered at its mean, is tight,
settling a long-standing conjecture. The proof combines a recent observation of
Bolthausen, Deuschel and Zeitouni with elements from (Bramson 1978) and
comparison theorems for Gaussian fields. An essential part of the argument is
the precise evaluation, up to an error of order 1, of the expected value of the
maximum of the GFF in a box. Related Gaussian fields, such as the GFF on a
two-dimensional torus, are also discussed.

Probability