  We characterise the existentially closed models of the theory of exponential
fields. They do not form an elementary class, but can be studied using positive
logic. We find the amalgamation bases and characterise the types over them. We
define a notion of independence and show that independent systems of higher
dimension can also be amalgamated. We extend some notions from classification
theory to positive logic and position the category of existentially closed
exponential fields in the stability hierarchy as NSOP$_1$ but TP$_2$.
