Breathers are temporally periodic, spatially localized, solutions of nonlinear wave equations or other, spatially extended, infinite dimensional dynamical systems. They are rare, because localized oscillations in such systems tend to be destroyed by resonances with the dispersive modes in the system. While breathers have been shown to exist some nonlinear wave equations with spatially periodic coefficients, these examples require rather special conditions on the coefficients. I'll explain why Nash-Moser, or "hard", implicit function theorems are a natural tool to try to expand the class of equations for which breathers exist.