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 NSOP1 but TP2.