When a virus infects a living cell it directs the biosynthetic resources of the cell toward transcription and translation of viral proteins, replication of viral genomes, assembly of virus particles, and release of hundreds to thousands of progeny virus particles to the extracellular environment. For well-characterized viruses one may begin to build kinetic models that predict virus growth behavior based on the dynamics of the underlying molecular processes within the infected cell. Recent experimental study of single BHK cells infected by single particles of vesicular stomatitis virus, a virus that encodes only five genes, reveal a broad distribution of virus productivity. Viral genetic variation and host-cell heterogeneity are unable to fully account for the broad growth behavior. A better understanding of these experiments may follow from stochastic modeling of virus growth.