Orderable groups in topology

In the last few years it has been realized that many of the groups that appear most naturally in low-dimensional topolgy are orderable. This means, their elements can be totally ordered in a way which is compatible with the group structure: g < h implies kg < kh for g,h,k in the group. Orderability is a stronger condition that torsion-freeness. First I want to explain the concept, and geometric significance of orderability. Then I want to talk about two classes of examples that I'm particularly interested in: (1) braid groups and their generalisations, and (2) fundamental groups of 3-manifolds. In both cases, I want to show how the problem of ordering these groups has shed new light on existing theory, and has sometimes revealed fascinating new problems.