Why is my dusters glove compartment full of water?

I'm guessing the windsheild gasket or the area behind the windsheild is rotted) or maybe a wiper pivot seal is bad. The water dam in the cowl ( as suggested) is not over the glove box, and when they do rot out they leak water into the heater box, sense thats what is below it. I would remove the glove box liner and investate with a flash light while someone sprays a garden hose at the windsheild for you. You will find the culprit.