The L-space conjecture relates non-left-orderability of 3-manifold groups to Heegaard Floer homology lens-spaces, or, L-spaces. In this talk I will give the definition of an L-space , and attempt to give a feeling for this class of manifolds by focusing on some examples. One class of examples is due to Ozsváth and Szabó: branched double covers of the 3-sphere with branch set a non-split alternating link. This leads to a surprising conjecture (implied by the L-space conjecture) relating simplicity in Khovanov homology to non-left-orderability of the fundamental group of the branched double cover.