In this application paper, we report on over fifteen years of experience with relativistic and astrophysical visualization, which has been culminating in a substantial engagement for visualization in the Einstein Year 2005 - the 100th anniversary of Einstein's publications on special relativity, the photoelectric effect, and Brownian motion. This paper focuses on explanatory and illustrative visualizations used to communicate aspects of the difficult theories of special and general relativity, their geometric structure, and of the related fields of cosmology and astrophysics. We discuss visualization strategies, motivated by physics education and didactics of mathematics, and describe what kind of visualization methods have proven to be useful for different types of media, such as still images in popular-science magazines, film contributions to TV shows, oral presentations, or interactive museum installations. Although our visualization tools build upon existing methods and implementations, these techniques have been improved by several novel technical contributions like image-based special relativistic rendering on GPUs, an extension of general relativistic ray tracing to manifolds described by multiple charts, GPU-based interactive visualization of gravitational light deflection, as well as planetary terrain rendering. The usefulness and effectiveness of our visualizations are demonstrated by reporting on experiences with, and feedback from, recipients of visualizations and collaborators.