  This is a draft of a book submitted for publication by the AMS. Its theme is
the remarkable interplay, accelerating in the last few decades, between
topology and the theory of orderable groups, with applications in both
directions.
  It begins with an introduction to orderable groups and their algebraic
properties. Many of the algebraic results are proved by topological methods,
via consideration of the space of orderings. After a discussion H\"older's
theorem and some dynamical aspects of orderable groups, we provide explicit
orderings of important groups in topology: free groups and most surface groups.
  Next we consider orderability of the fundamental groups of three-dimensional
manifolds. All knot groups, and more generally groups of 3-manifolds with
positive first Betti number are left-orderable, in fact locally indicable and
sometimes even bi-orderable. However when the first homology is finite the
situation is more subtle, and we find connections with foliations, branched
coverings, surgery, and conjecturally Heegaard-Floer homology.
  Braid groups are considered in some detail, including Dehornoy's ordering of
the braid groups and its later generalizations due to Thurston. This is
followed by a discussion of recent applications of Dehornoy's ordering to knot
theory. A short chapter outlines a proof that the group of PL homeomorphisms of
the $n$-dimensional cube (fixed on the boundary) is left-orderable; a property
conjectured to be true for the group of homeomorphisms in dimension two.
  We present a new proof that local indicability of a group is equivalent to
the existence of a "Conradian" left-ordering. A final chapter considers the
space $LO(G)$ of left-orderings of a group $G$. We give a new proof of Sikora's
theorem that $LO(\mathbb{Z}^n), n >1$ is a Cantor set, as well as a proof of
Linnell's theorem that for any group $G$, $LO(G)$ is either finite or
uncountable.
