  The strict connection between Lie point-symmetries of a dynamical system and
its constants of motion is discussed and emphasized, through old and new
results. It is shown in particular how the knowledge of a symmetry of a
dynamical system can allow to obtain conserved quantities which are invariant
under the symmetry. In the case of Hamiltonian dynamical systems it is shown
that, if the system admits a symmetry of "weaker" type (specifically, a
\lambda\ or a \Lambda-symmetry), then the generating function of the symmetry
is not a conserved quantity, but the deviation from the exact conservation is
"controlled" in a well defined way. Several examples illustrate the various
aspects.
