Why don't my pumpkins have any pumpkins?

Pumpkin plants are easy to grow and popular with many food gardeners.  However, it is quite common for people to become somewhat worried as the season progresses.  'My pumpkin plants have gone berserk.  They now take up this really large area, they look incredibly healthy, I have seen many flowers, but I don't see many pumpkins.  What's wrong?'