If we were to have a Grid infrastructure for visualization, what technologies would be needed to build such an infrastructure, what kind of applications would benefit from it, and what challenges are we facing in order to accomplish this goal? In this report, we make use of the term `visual supercomputing' to encapsulate a subject domain concerning the infrastructural technology for visualization. We consider a broad range of scientific and technological advances in computer graphics and visualization, which are relevant to visual supercomputing. We identify the state of the art technologies that have prepared us for building such an infrastructure. We examine a collection of applications that would benefit enormously from such an infrastructure, and discuss their technical requirements. We propose a set of challenges that may guide our strategic efforts in the coming years.