We discuss two methods for constraining the equation of state of dark energy using strong gravitational lensing by galaxy clusters. In the so called “arc statistics” approach, we compare the cluster efficiency for producing giant arcs in several dark-energy cosmologies and in the “standard” $\Lambda$CDM and OCDM models. We find that the expected abundance of gravitational arcs depends on the equation of state of dark energy and reflects the dependence of halo concentrations on cosmology. In agreement with results in previous works, the lensing cross section is very sensitive to dynamical processes occurring in the lenses. Then we use gravitational arcs for tracing the position of the lens critical curves and we measure their scaling with the source redshift in a variety of cosmological models. We find that there is a degeneracy between several lens properties and the equation of state of dark energy which can be broken only after an extremely precise modeling of the lens. Instead of using this “golden lens” approach, we check whether combining the information from a statistical sample of clusters we can distinguish among the various cosmologies. We test the method on a sample of numerically simulated clusters and we reproduce the results expected from the analytic models.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html