I have heard people discussing the utility of $L$-functions, claiming that since they are essentially cohomological entities, they are "abelian" and therefore lack force.

From looking around on the web, I see that this idea has a base of believers, and that there is some notion of non-abelian $L$-functions.

### Question

What is the definition of non-abelian $L$-functions? Does it have to do with replacing cohomology with homotopy in some way? How does it relate to the original definition of $L$-function (in particular, what is the analogue of the characteristic polynomial?)? What is the context in which it arises?