There are a number of specialties in low-dimensional topology that can find in their 'family tree' a common ancestry in the theory of surface mappings. These include knot theory as studied through the use of braid representations and 3-manifolds as studied through the use of Heegaard splittings. The study of the surface mapping class group (the modular group) is of course a rich subject in its own right, with relations to many different fields of mathematics and theoretical physics. But its most direct and remarkable manifestation is probably in the vast area of low-dimensional topology. Although the scene of this area has been changed dramatically and experienced significant expansion since the original publication of Professor Joan Birman's seminal work, "Braids, Links, and Mapping Class Groups" (Princeton University Press), she brought together mathematicians whose research span many specialities, all of common lineage.The topics covered are quite diverse. Yet they reflect well the aim and spirit of the conference in low-dimensional topology held in honor of Joan S.Birman's 70th birthday at Columbia University (New York, NY), which was to explore how these various specialties in low-dimensional topology have diverged in the past 20-25 years, as well as to explore common threads and potential future directions of development.