  In this paper we consider the structure of $\omega$-limit sets in subshifts
of Baire space. We consider both subshifts of finite type and subshifts of
bounded type and we demonstrate that many classical structure theorems for
$\omega$-limit sets fail in this context. Nevertheless, we obtain
characterizations of $\omega$-limit sets in subshift of finite types and of
attracting $\omega$-limit sets in subshifts of bounded type.
