In the early 90's, physicists Bershadsky-Cecotti-Ooguri-Vafa conjectured that the analytic torsion was the counterpart in complex geometry of the counting problem of elliptic curves in Calabi-Yau threefolds. It seems that this conjecture is not as well known as the usual mirror symmetry conjecture on the counting of rational curves. In this talk, I will explain the BCOV conjecture and some of its expected consequences. If time permits, I will also explain the construction of an analytic torsion for Calabi-Yau orbifolds and an explicit formula as a function on the moduli space.