Abstract WeidentifythesymmetryalgebraoftheLaplacianonEuclideanspaceas anexplicitquotient of the universalenvelopingalgebraofthe Liealgebraof conformalmotions.Weconstructanaloguesofthesesymmetriesonageneral conformalmanifold. 1.Introduction 2.Notationandstatement of results 3.Resultsin the flat case 4.The algebraic structure of An 5.Explicit formulae and the curved case 6.Concluding remarks