What are the applications (physical and mathematical) of classical field theory beyond electrodynamics and gravity?

By such applications, I mean that either the field theory viewpoint adds some genuinely new insight into the underlying physics or that it gives rise to interesting mathematical problems. So I'm not thinking about:

-field-theoretical description of something that is very well understood with other tools (for example, describing classical electrodynamics in language of fibre bundles, differential forms etc. is very nice and elegant, but doesn't add much to physics)

-quantum field theory (in QFT you always write down the classical lagrangian and then turn the fields into operators, but there is not much actual classical field theory here)

Of course, you can always write some lagrangian like phi^34 + 14*phi^8 + ..., and study the resulting PDE (existence and uniqueness of solutions etc.), but I guess that lacks real motivation.