The mathematical study of knots has a long and fascinating history and is currently a very lively area of research and application. I will discuss one of the earliest and most powerful algebraic invariants of knots: the fundamental group of its complement, commonly called the knot’s group. Although it is routine to find generators and relations for its group from a picture of a given knot, the problem of whether two presentations actually define isomorphic groups can be quite intractable. On the positive side, and rather surprisingly, all knot groups are left-orderable. This means that the elements of the group can be given a strict total ordering which is invariant under left multiplication (or right multiplication, if one prefers, but not necessarily both simultaneously). However, the groups of some knots, such as the figure-eight knot, can be given a two-sided ordering. These facts give new algebraic information about knot groups and have consequences regarding surgery and similar application of knot theory to other branches of topology. I’ll conclude by discussing some open questions, such as a conjecture relating left-orderability with Heegaard-Floer homology of 3-dimensional manifolds.