Title: L-spaces and left-orderability.

Abstract: A group is left-oderable if it admits a strict total order of its elements that is invariant under multiplication on the left; an L-space is a rational homology sphere with simplest possible Heegaard Floer homology. It has been conjectured that an irreducible rational homology sphere is an L-space if and only if it has non-left-orderable fundamental group. While this conjecture seems very optimistic, I will discuss some of the evidence for it. This will centre around joint projects with S. Boyer and C. Gordon and with A. Clay.