I will briefly review basic features of operadic categories that were introduced by Batanin and Markl as an attempt for general theory of "operad-like" structures. Regarding examples, I will focus on the operadic category governing modular operads.
In the second part of my talk I will recall Grothendieck's construction for operadic categories and explain how it can be used to construct new operadic categories from the old ones. The examples here will be operadic categories describing various operad- and PROP-like structures.