The classification of irreducible representations of finite and compact Lie groups over $\mathbb{C}$

Question

I just started reading a few lecture notes on representation theory and I was wondering about a big picture that one should keep in mind while reading through these lecture notes.

Have all (finite-dimensional) irreducible representations of finite groups and compact Lie groups over $\mathbb{C}$ been classified in some way?

I am guessing that all (finite-dimensional) irreducible representations of a general Lie group haven't yet been classified, or have they?

Thank you.