We use the term Unit Visualizations to describe a class of visualizations that explicitly represent every row in a data set. They have been around in one form or another for hundreds of years, usually in static form (e.g. tallies, scatterplots, Dot Plots, Unit Charts, Pixel Charts, or Isotypes.) We characterize their design space and propose a unifying framework that can produce common types of Unit Visualizations. In addition, we introduce SandDance, a tool built to explore the effectiveness of animating between and interacting with unit representations when analyzing and presenting multidimensional datasets.