This monograph contains three lecture series from the SMF school ``Geometric and quantum topology in dimension $3$'', which was held at CIRM in June 2014. These lectures present recent progress on the study of $3$-manifold and link invariants.Thang Lê describes the state of the art about the AJ conjecture, which relates generalizations of the Jones polynomial to the Cooper, Culler, Gillet, Long and Shalen $A$-polynomial, which is defined from $SL_2(C)$-representation spaces of link exterior fundamental groups.In 1999, Khovanov defined a homology theory for knots of $R^3$ whose Euler characteristic is the Jones polynomial. Paul Turner presents the latest developments and the applications of this categorification of the Jones polynomial in a useful guide of the literature around this extensively studied topic.Robert Lipshitz presents the famous Osv\'ath Sz\'abo Heegaard Floer homology theories together with efficient sketches of proofs of some of their spectacular applications.These lectures are introduced by a partial survey of the history of these invariants, written by Christine Lescop.