Abstract: The L-space conjecture connects topological (taut foliations), analytic (Floer invariants), and algebraic (orderability of groups) properties of $3$-manifolds, which has become a popular topic in low-dimensional topology. In this talk, we will first give an overview of the conjecture, including its motivation, known results, and related open questions. Then in the second part of the talk, we will discuss my joint works with Steve Boyer and Cameron Gordon on slope detections and their applications to the L-space conjecture for toroidal 3-manifolds.