Directed, purposeful movement is one of the qualities that we most closely associate with living organisms, and essentially all known forms of life on this planet exhibit some type of self-generated movement or motility. Even organisms that remain sessile most of the time, like flowering plants and trees, are quite busy at the cellular level, with large organelles, including chloroplasts, constantly racing around within cellular boundaries. Directed biological movement requires that the cell be able to convert its abundant stores of chemical energy into mechanical energy. Understanding how this mechanochemical energy transduction takes place and understanding how small biological forces generated at the molecular level are marshaled and organized for large-scale cellular or organismal movements are the focus of the field of cell motility. This tutorial, aimed at readers with a background in physical sciences, surveys the state of current knowledge and recent advances in modeling cell motility.