Fatou components for rational functions in the Riemann sphere are very well understood and play an important role in our understanding of one-dimensional dynamics. In higher dimensions the situation is less well understood. In this work we give a classification of invariant Fatou components for moderately dissipative Hénon maps. Most of our methods apply in a much more general setting. In particular we obtain a partial classification of invariant Fatou components for holomorphic endomorphisms of projective space, and we generalize Fatou's Snail Lemma to higher dimensions.