Since its introduction in 1987, the incredibly beautiful geometry of
Hitchin fibrations have had deep impact on many branches of
mathematics, from mathematical physics, symplectic geometry, to geometric
representation theory, number theory, and more. In recent years, there has been
a lot of progress towards undertanding its non-linear analogues, which have shown
to possess even richer geometry and possibly even bigger potential in
applications. I will give an introductory overview of
such generalized Hitchin systems and highlight some recent developments and
potential future directions about its geometry. The emphasis will be on
stability questions related to multiplicative Higgs bundles, joint work in
progress with Siqing Zhang and Andres Fernandez Herrero.