This book is an introduction to combinatorial torsions of cellular spa ces and manifolds with special emphasis on torsions of 3-dimensional m anifolds. The first two chapters cover algebraic foundations of the th eory of torsions and various topological constructions of torsions due to K. Reidemeister, J.H.C. Whitehead, J. Milnor and the author. We al so discuss connections between the torsions and the Alexander polynomi als of links and 3-manifolds. The third (and last) chapter of the book deals with so-called refined torsions and the related additional stru ctures on manifolds, specifically homological orientations and Euler s tructures. As an application, a construction of the multivariable Conw ay polynomial of links in homology 3-spheres is given. At the end of t he book, the recent results of G. Meng, C.H. Taubes and the author on the connections between the refined torsions and the Seiberg-Witten in variant of 3-manifolds are briefly described.