Scheinerman and Wilf SW94] assert that \an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph K n ." A rectilinear embedding or drawing of K n is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear. The rectilinear crossing number of K n is the fewest number of edge crossings attainable over all planar rectilinear embeddings of K n. For each n we construct a rectilinear embedding of K n that has the fewest number of edge crossings and the best asymptotics known to date. Moreover, we give some alternative innnite families of embeddings of K n with good asymptotics. Finally, we mention some old and new open problems.