The Most Confusing Word

Prego

1] Spoken by a waiter with a grand sweeping gesture inviting us to sit and have lunch at his café.

2] Spoken by an angry museum curator, just before she shooes people who are standing too close to old statues.

3] Spoken by the hotel concierge in response to our “graztie.”

The last scenario actually made sense to us, because according to our little Italian phrase book, prego means you're welcome. But given all the other situations, we were left a bit befuddled.

So what, pray tell, does that word mean?!