  We report the first results of a spectroscopic survey of galaxies in the core
of the Shapley Concentration, the richest nearby supercluster of clusters of
galaxies. We have measured 311 new galaxy redshifts in an area of $\sim 4.5$
square degrees centered around the Abell cluster A3558. Considering also the
data already available in the literature, the total number of galaxy redshifts
in this area amounts to more than 500. On the basis of these data we estimate
the mean velocities and the velocity dispersions of the Abell clusters A3556,
A3558 and the poor cluster SC 1329 -314. Finally, from an analysis of the
projected and three--dimensional distributions of galaxies in this region, we
estimate the galaxy overdensity and find that the core of the Shapley
Concentration has an interesting, very complex dynamical state: the main
clusters appear to be interacting with each other, forming a single elongated
structure containing many subcondensations.
