In the present paper, a survey about some results from the theory of exponential fields is given. Then investigations are motived by Tarski's decidability problem of the field of real numbers with an additional exponential function. For solving Tarski's problem it seems to be useful to have more information about special exponential fields and es of such structures. So different axiomatic es of exponential fields and their theories are investigated. Especially, the solution of the “dominance problem” and the “problem of the last root” for exponential terms are given here.