The standard model of particle physics is based on the gauge group $U(1) \times SU(2) \times SU(3)$ and describes all well-known physical interactions but with exception that gravity isn't involved. And even gravity (in classical theory) has a gauge group: Local translations in spacetime.
Why every fundamental interaction that we can observe is written in terms of a gauge-invariant Lagrangian?
What is in the case if an experiment would make evidence that there is another new fundamental interaction between particles or particle systems; would it be also modelled with a gauge theory?
Another interesting example is the Kaluza-Klein theory: Here, General Relativity is extended to 5 dimensions with one dimension compactified to a circle; the result is surprisingly: Einstein's General relativity (for gravity) + Electromagnetism. Later there were physicist who tried to define gauge theory by extending General relativity to more dimensions and compute the Ricci scalar (historically, this procedure was used before nonabelian gauge theories were found).
What is the mathematical background between the Kaluza-Klein theory?
Can a program similar to Kaluza and Klein help someone to define a more general gauge theory than Yang-Mills nonabelian gauge theories?
Or is the Yang-Mills theory (I am assuming no supersymmetry!) the most general gauge theory?