  We consider the problem of computing the overlaps between the Bethe states of
the XXZ spin-1/2 chain and generic states. We derive recursive formulas for the
overlaps between some simple product states and off-shell Bethe states within
the framework of the Algebraic Bethe Ansatz. These recursive formulas can be
used to prove in a simple and straightforward way the recently-obtained results
for the overlaps of the Bethe states with the N\'eel state, the dimer state,
and the \textit{q}-deformed dimer state. However, these recursive formulas are
derived for a broader class of states and represent a concrete starting point
for the computation of rather general overlaps. Our approach can be easily
extended to other one-dimensional Bethe Ansatz integrable models.
