An Exponential History of Functions with Logarithmic Growth

We survey recent work on normal functions, including limits and singularities of admissible normal functions, the Griffiths-Green approach to the Hodge conjecture, algebraicity of the zero-locus of a normal function, Neron models, and Mumford-Tate groups. Some of the material and many of the examples, esp. in sec. 5-6, are original. The first two sections should be easily accessible to graduate students.