To follow quickly, often in an effort to catch or catch up with (someone or something).
That dog will get hurt if he continues to run after cars.
To make a determined effort to win someone's affections or to have a sexual relationship with them.
She runs after any man in uniform!
To endeavour to find or obtain.
To seek the company or assistance of; to frequent; to consult (someone).
To seek to participate in, devote oneself to (something).