Non-perturbative physics

Could we ever be able to a get a handle on non-perturbative physics?
Certainly there is a lot of mathematical obscurities on the way.
Any achievement on this front can bring up a profound insight into our understanding of physical systems.
Honestly, how much do we know about strongly correlated systems?

To my physicist friends: If you've any ideas or comments, I'd be glad if you share them with me.