TITLE: Forbidden toroidal minors
QUESTION [5 upvotes]: A finite graph is planar if and only if it does not have $K_5$ or $K_{3,3}$ as a minor. Is there a (finite) set of minors that can classify if a graph is toroidal?

REPLY [5 votes]: Yes, see Robertson-Seymour theorem. Additionally

The complete set of forbidden minors for toroidal graphs remains unknown, but contains at least 16000 graphs.