  Motivated by a problem on the dynamics of compositions of plane hyperbolic
isometries, we prove several fundamental results on semigroups of isometries,
thought of as real M\"obius transformations. We define a semigroup $S$ of
M\"obius transformations to be \emph{semidiscrete} if the identity
transformation is not an accumulation point of $S$. We say that $S$ is
\emph{inverse free} if it does not contain the identity element. One of our
main results states that if $S$ is a semigroup generated by some finite
collection $\mathcal{F}$ of M\"obius transformations, then $S$ is semidiscrete
and inverse free if and only if every sequence of the form $F_n=f_1\dotsb f_n$,
where $f_n\in\mathcal{F}$, converges pointwise on the upper half-plane to a
point on the ideal boundary, where convergence is with respect to the chordal
metric on the extended complex plane. We fully classify all two-generator
semidiscrete semigroups, and include a version of J{\o}rgensen's inequality for
semigroups.
  We also prove theorems that have familiar counterparts in the theory of
Fuchsian groups. For instance, we prove that every semigroup is one of four
standard types: elementary, semidiscrete, dense in the M\"obius group, or
composed of transformations that fix some nontrivial subinterval of the
extended real line. As a consequence of this theorem, we prove that, with
certain minor exceptions, a finitely-generated semigroup $S$ is semidiscrete if
and only if every two-generator semigroup contained in $S$ is semidiscrete.
  After this we examine the relationship between the size of the `group part'
of a semigroup and the intersection of its forward and backward limit sets. In
particular, we prove that if $S$ is a finitely-generated nonelementary
semigroup, then $S$ is a group if and only if its two limit sets are equal.
  We finish by applying some of our methods to address an open question of
Yoccoz.
