Grid computing provides a challenge for visualization system designers. In this research, we evolve the dataflow concept to allow parts of the visualization process to be executed remotely in a secure and seamless manner. We see dataflow at three levels: an abstract specification of the intent of the visualization; a binding of these abstract modules to a specific software system; and then a binding of software to processing and other resources. We develop an XML application capable of describing visualization at the three levels. To complement this, we have implemented an extension to a popular visualization system, IRIS Explorer, which allows modules in a dataflow pipeline to run on a set of grid resources. For computational steering applications, we have developed a library that allows a visualization system front-end to connect to a simulation running remotely on a grid resource. We demonstrate the work in two applications: the dispersion of a pollutant under different wind conditions; and the solution of a challenging numerical problem in elastohydrodynamic lubrication.