We probe gravitational clustering in N-body simulations using geometrical descriptors sensitive to ‘connectedness’: the genus curve, percolation and shape statistics. As gravitational clustering advances, the density field in N-body simulations shows an increasingly pronounced departure from Gaussianity reflected in the changing shape of the percolation curve and the changing amplitude and shape of the genus curve. We feel that both genus and percolation curves provide complementary probes of large scale structure topology and could be used to discriminate between models of structure formation and the analysis of observational data such as galaxy catalogs and MBR maps. The filling factor in clusters & superclusters at percolation is small indicating that matter is more likely to lie in filaments and pancakes. An analysis of ‘shapes’ in N-body simulations has shown that filaments are more pronounced than pancakes. To probe shapes of clusters and superclusters more rigorously we propose a new shape statistic which does not fit isodensity surfaces by ellipsoids (as done earlier). Instead our shape statistic is derived from fundamental properties of a compact body such as its volume V, surface area S, integrated mean curvature C, and connectivity (characterized by the Genus). The new shape statistic gives sensible results for topologically simple surfaces such as the ellipsoid, and for more complicated surfaces such as the torus.