This work is a detailed version of arXiv:0704.0195 [math.QA].
We introduce a new notion of the core of a braided fusion category. It allows
to separate the part of a braided fusion category that does not come from
finite groups.
We also give a comprehensive and self-contained exposition of the known
results on braided fusion categories without assuming them pre-modular or
non-degenerate.
The guiding heuristic principle of our work is an analogy between braided
fusion categories and Casimir Lie algebras.
