  We present theoretical and observational studies of strong gravitational
lenses produced by clusters of galaxies. Our purpose is to test the Cold Dark
Matter (CDM) model at small and highly non-linear scales where it has been
claimed that the CDM model may confront several difficulties. We concentrate
our attention on the statistics of strong gravitational lenses. We use two
complementary statistics, lensed arcs and quasars, to probe the mass
distributions. First, we construct a triaxial lens model, and develop a new
method to include triaxiality of dark halos in the lens statistics. We find
that the effect of triaxiality is significant; it enhances lensing
probabilities by a factors of a few to ten, assuming the degree of triaxiality
predicted in the CDM model. In particular, we argue that both central
concentration and large triaxiality of dark halos are required to reproduce the
observed number of arcs in clusters; thus the result can be interpreted as a
strong evidence for the cold and collisionless dark matter. One of the most
notable advantages of the triaxial modeling is that the triaxial modeling
allows us to predict image multiplicities. We find that the CDM halos predict
significant fraction (more than 20%) of naked cusp lenses. In addition, we
point out the image multiplicities depend strongly on the central concentration
of dark halos. Therefore we propose image multiplicities as a new powerful test
of the CDM model. Finally we searched for large-separation lensed quasars from
the data of the Sloan Digital Sky Survey, and succeeded in discovering the
first large-separation lensed quasar SDSS J1004+4112. We find that the
discovery of the large-separation quadruple lens SDSS J1004+4112 is quite
consistent with the theoretical predictions based on the CDM model. (Abridged)
