  The algebra of exponential fields and their extensions is developed. The
focus is on ELA-fields, which are algebraically closed with a surjective
exponential map. In this context, finitely presented extensions are defined, it
is shown that finitely generated strong extensions are finitely presented, and
these extensions are classified. An algebraic construction is given of Zilber's
pseudo-exponential fields. As applications of the general results and methods
of the paper, it is shown that Zilber's fields are not model-complete,
answering a question of Macintyre, and a precise statement is given explaining
how Schanuel's conjecture answers all transcendence questions about
exponentials and logarithms. Connections with the Kontsevich-Zagier,
Grothendieck, and Andr\'e transcendence conjectures on periods are discussed,
and finally some open problems are suggested.
