  We introduce and study \emph{shift-similar} groups
$G\le\textrm{Sym}(\mathbb{N})$, which play an analogous role in the world of
Houghton groups that self-similar groups play in the world of Thompson groups.
We also introduce Houghton-like groups $H_n(G)$ arising from shift-similar
groups $G$, which are an analog of R\"over-Nekrashevych groups from the world
of Thompson groups. We prove a variety of results about shift-similar groups
and these Houghton-like groups, including results about finite generation and
amenability. One prominent result is that every finitely generated group embeds
as a subgroup of a finitely generated shift-similar group, in contrast to
self-similar groups, where this is not the case. This establishes in particular
that there exist uncountably many isomorphism classes of finitely generated
shift-similar groups, again in contrast to the self-similar situation.
