Over the past quarter century, light-source user facilities have transformed research in areas ranging from gas-phase chemical dynamics to materials characterization. The ever-improving capabilities of these facilities have revolutionized our ability to study the electronic structure and dynamics of atoms, molecules, and even the most complex new materials, to understand catalytic reactions, to visualize magnetic domains, and to solve protein structures. Yet these outstanding facilities still have limitations well understood by their thousands of users. Accordingly, over the past several years, many proposals and conceptual designs for "next-generation" x-ray light sources have been developed around the world. In order to survey the scientific problems that might be addressed specifically by those new light sources operating below a photon energy of about 3 keV and to identify the scientific requirements that should drive the design of such facilities, a workshop "Science for a New Class of Soft X-Ray Light Sources" was held in Berkeley in October 2007. From an analysis of the most compelling scientific questions that could be identified and the experimental requirements for answering them, we set out to define, without regard to the specific technologies upon which they might be based, the capabilities such light sources would have to deliver in order to dramatically advance the state of research in the areas represented in the programs of the Department of Energy's Office of Basic Energy Sciences (BES). This report is based on the workshop presentations and discussions.