This lecture notes is based on a sequence of lectures on Reidemeister torsions by V Turaev.J Milnor showed that Reidemeister torsions and the Alexander polynomials are deeply related. By the surgery formulas of Reidemeister torsions, we can apply them to Dehn surgery problems, namely to lens and Seifert surgery problems, and to amphicheirality problem of a link. Since Reidemeister torsions of rational homology 3-spheres are elements in cyclotomic fields, we need algebraic number theoretical studies.Recently P Ozsvath and Z Szabo defined the Heegaard Floer homology which is a powerful invariant to study 3-manifolds. It induces the Reidemeister-Turaev torsion. We also explain it here.