Although R includes numerous tools for creating color palettes to display continuous data, facilities for displaying categorical data primarily use the RColorBrewer package, which is, by default, limited to 12 colors. The colorspace package can produce more colors, but it is not immediately clear how to use it to produce colors that can be reliably distingushed in different kinds of plots. However, applications to genomics would be enhanced by the ability to display at least the 24 human chromosomes in distinct colors, as is common in technologies like spectral karyotyping. In this article, we describe the Polychrome package, which can be used to construct palettes with at least 24 colors that can be distinguished by most people with normal color vision. Polychrome includes a variety of visualization methods allowing users to evaluate the proposed palettes. In addition, we review the history of attempts to construct qualitative color palettes with many colors.