Is it possible for a woman to be born barren?

I was watching a movie about a woman who only had one egg left and after she was supposed to have her period, she would lose it. It got me thinking about how you are born with a set number of eggs and you go through menopause after that number is released. So just wondering but is it possible for a girl to be born with no eggs?Another question I have is if a woman becomes barren, does she still have a period?