  We prove various results that, given a sufficiently rich subgroup $G$ of the
group of homeomorphisms on the real line, describe the structure of the other
possible actions of $G$ on the line, and address under which conditions such
actions must be semi-conjugate to the natural defining action of $G$. The main
assumption is that $G$ should be locally moving, meaning that for every open
interval the subgroup of elements fixing pointwise its complement, acts on it
without fixed points. We show that when $G$ is a locally moving group, every
$C^1$ action of $G$ on the real line is semi-conjugate to its standard action
or to a non-faithful action. The situation is much wilder when considering
actions by homeomorphisms: for a large class of groups, we describe uncountably
many conjugacy classes of faithful minimal actions. Next, we prove structure
theorems for $C^0$ actions, based on the study of laminar actions, which are
actions on the line preserving a lamination. When $G$ is a group of
homeomorphisms of the line acting minimally, and with a non-trivial compactly
supported element, then any faithful minimal action of $G$ on the line is
either laminar or conjugate to its standard action. Moreover, when $G$ is a
locally moving group with a suitable finite generation condition, for any
faithful minimal laminar action there is a map from the lamination to the line,
called a horograding, which is equivariant with respect to the action on the
lamination and the standard one, and with some extra suitable conditions. This
establishes a tight relation between all minimal actions on the line of such
groups, and their standard actions. Finally, based on an analysis of the space
of harmonic actions, we show that for a large class of locally moving groups,
the standard action is locally rigid, in the sense that sufficiently small
perturbations in the compact-open topology give semi-conjugate actions.
