The electron collisional damping of various kinds of Alfven eigenmodes (AE) in stellarators is studied. Simple expressions for the damping rates of various well-localized AEs are obtained and applied to evaluating the damping rates in the Helias reactor [H. Wobig , Fusion Energy 1998, 17th Conference Proceedings, Yokohama, Japan (IAEA, Vienna, 1999), Vol. 4, p. 1235] and quasi-poloidal stellarator [J. F. Lyon , Fusion Energy 2002, 19th Conference Proceedings, Lyon, France (IAEA, Vienna, 2002), Paper IC/P-05]. The estimates made indicate that the collisional damping prevents the destabilization of AEs localized at the plasma periphery of the Helias reactor. A new mechanism that can considerably affect the collisional damping of Alfven waves in stellarators is predicted: it is found that under certain conditions the collisionless orbit transformations between the locally trapped states and locally passing states significantly change the damping rate, so that the latter becomes weakly (logarithmically) dependent on the collision frequency. (C) 2004 American Institute of Physics.