  We develop an abstract model for the dynamics of an exponential map $z\mapsto
\exp(z)+\kappa$ on its set of escaping points and, as an analog of Boettcher's
theorem for polynomials, show that every exponential map is conjugate, on a
suitable subset of its set of escaping points, to a restriction of this model
dynamics. Furthermore, we show that any two attracting and parabolic
exponential maps are conjugate on their sets of escaping points; in fact, we
construct an analog of Douady's "pinched disk model" for the Julia sets of
these maps. On the other hand, we show that two exponential maps are generally
not conjugate on their sets of escaping sets. Using the correspondence with our
model, we also answer several questions about escaping endpoints of external
rays, such as when a ray is differentiable in such an endpoints or how slowly
these endpoints can escape to infinity.
