The purpose of this paper is to describe Lagrangian Mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard tangent bundles...). In particular, we are interested in two cases: singular Lagrangian systems and vakonomic mechanics (variational constrained mechanics). Several examples illustrate the interest of these developments.This work has been partially supported by MEC (Spain) Grants MTM 2006-03322, MTM 2004-7832, project “Ingenio Mathematica” (i-MATH) No. CSD 2006-00032 (Consolider-Ingenio 2010) and S-0505/ESP/0158 of the CAM. D. Iglesias wants to thank MEC for a Research Contract "Juan de la Cierva".Peer reviewed