In this talk, I will review recent advances in solving this problem using index calculus algorithms, starting from the work of Semaev in 2004. As it happens, we now have subexponential (in L(2/3) time) algorithms for special families of parameters, but these parameters are however not really used in practice. I will then show how these algorithms can potentially be adapted to elliptic curves defined over binary fields of prime degree extensions and to elliptic curve defined over prime fields (the two families that appear in standards and applications), and I will describe remaining challenges in improving both their complexity and their analysis.