The L-space conjecture asserts the equivalence of three properties of a prime 3-manifold M: (1) the fundamental group of M has a left-invariant total order, (2) M admits a taut 2-dimensional foliation, and (3) the Heegaard Floer homology of M is "large". One of the interests of this conjecture is that these properties have very different flavors: (1) is algebraic, (2) is topological, and (3) is essentially analytic.

We will define the three properties and describe what is known about the conjecture. In particular we will discuss some recent joint work with Steve Boyer and Ying Hu on the case of toroidal 3-manifolds.