Many critical biological processes involve the uni-directional transport of particles along one-dimensional structures. Examples of such processes include ribosome motion along mRNA, the movement of polymerases along DNA, and kinesin motion along microtubules. A simple mathematical model that captures the salient properties of these biological transport processes is the totally asymmetric exclusion model (TASEM). We will introduce the TASEM, discuss well-known approaches to the model, and examine the nature of the solutions. Additionally, we will introduce recently published work that extends the model to address the transport of finite-size particles. Finally, ongoing efforts to extend the model to the case of more complex transport behaviors will be described.