check, check into, check out, check over, check up on, suss out, go over, look into - examine so as to determine accuracy, quality, or condition; "check the brakes"; "Check out the engine"

trace, follow - follow, discover, or ascertain the course of development of something; "We must follow closely the economic development is Cuba" ; "trace the student's progress"

keep tabs on - keep a record on or watch attentively; "The government keeps tabs on the dissidents"

guard - to keep watch over; "there would be men guarding the horses"

invigilate, proctor - as of students taking an exam, to prevent cheating