Automata theory aficionado here. Regular languages are really neat! They have an air of being an "important" mathematical object, given how many different "natural" characterizations of them exist.

In a standard CS course you'll start out learning that regular expressions and finite state automata define the same class of languages. But they're also equivalent to:

- Languages generated by regular grammars (only rules of the form A -> a, A -> Ba, or A -> epsilon)

- Languages definable in monadic second-order (MSO) logic

- Languages "recognized" by a finite monoid (this algebraic appraoch to formal languages is super interesting and rich!)

- Language's whose Myhill-Nerode relation has finitely many equivalence classes