  In this paper we introduce a new class of domains in complex Euclidean space,
called Goldilocks domains, and study their complex geometry. These domains are
defined in terms of a lower bound on how fast the Kobayashi metric grows and an
upper bound on how fast the Kobayashi distance grows as one approaches the
boundary. Strongly pseudoconvex domains and weakly pseudoconvex domains of
finite type always satisfy this Goldilocks condition, but we also present
families of Goldilocks domains that have low boundary regularity or have
boundary points of infinite type. We will show that the Kobayashi metric on
these domains behaves, in some sense, like a negatively curved Riemannian
metric. In particular, it satisfies a visibility condition in the sense of
Eberlein and O'Neill. This behavior allows us to prove a variety of results
concerning boundary extension of maps and to establish Wolff-Denjoy theorems
for a wide collection of domains.
