In reply to Tall Clare:
Jane Bown's portrait of Samuel Beckett is one of the best portrait photos I can think of.
I can also remember going to an exhibition of Yousef Karsh portraits, and falling in love with his images of Audrey Hepburn.
Karsh also shows that a portrait doesn't have to be a traditional facial image - he has a portrait of the cellist Pablo Casals, shot from behind so his face isn't visible, but he's playing his cello and the image captures his personality very effectively.