  The dynamics of transcendental functions in the complex plane has received a
significant amount of attention. In particular much is known about the
description of Fatou components. Besides the types of periodic Fatou components
that can occur for polynomials, there also exist so-called Baker domains,
periodic components where all orbits converge to infinity, as well as wandering
domains.
  In trying to find analogues of these one dimensional results, it is not clear
which higher dimensional transcendental maps to consider. In this paper we find
inspiration from the extensive work on the dynamics of complex H\'enon maps. We
introduce the family of transcendental H\'enon maps, and study their dynamics,
emphasizing the description of Fatou components. We prove that the
classification of the recurrent invariant Fatou components is similar to that
of polynomial H\'enon maps, and we give examples of Baker domains and wandering
domains.
