  In this paper we study when the Kobayashi distance on a Kobayashi hyperbolic
domain has certain visibility properties, with a focus on unbounded domains.
"Visibility" in this context is reminiscent of visibility, seen in negatively
curved Riemannian manifolds, in the sense of Eberlein-O'Neill. However, we do
not assume that the domains studied are Cauchy-complete with respect to the
Kobayashi distance, as this is hard to establish for domains in $\mathbb{C}^d$,
$d \geq 2$. We study the various ways in which this property controls the
boundary behavior of holomorphic maps. Among these results is a
Carath\'eodory-type extension theorem for biholomorphisms between planar
domains, notably: between infinitely-connected domains. We also explore
connections between our visibility property and Gromov hyperbolicity of the
Kobayashi distance.
