The Reidemeister torsion is historically the first invariant of a manifold which is not preserved by homotopy. In 1971, Ray and Singer proposed an analytic analogue of the Reidemeister torsion, and conjectured that the two invariants are equal. This conjecture was proven 7 years later in the celebrated papers by Cheeger and Muller.
In the talk I will review the constructions of the Reidemeister and the Ray-Singer torsions and also will present a new invariant, the refined analytic torsion, recently introduced by T.Kappeler and myself. The refined analytic torsion is a holomorphic function on the space of representations of the fundamental group of a closed odd-dimensional manifold, whose absolute value is equal to the Ray-Singer torsion and whose phase is related to the Atiyah-Patody-Singer eta-invariant. The fact that the Ray-Singer torsion and the eta-invariant can be combined into one holomorphic function allows to use methods of complex analysis to study both invariants. I will present some applications of this method. In particular, I will prove a refinement of the Ray-Singer conjecture, establishing the relationship between the refined analytic torsion and the Turaev's refinement of the Reidemeister torsion. Another application is a calculation of so called spectral flow using the methods of combinatorial topology. I will also discuss the relationship of our construction with other recent constructions of the complex-valued torsions due to Burghelea-Haller and Cappell-Miller.