Many critical biological transport processes occur along one dimensional pathways. Examples include the translocation of RNA and DNA polymerases, and ribosome motion along mRNA. A model which captures many of the salient features of these biological transport processes is the totally asymmetric simple exclusion process (TASEP). We will review the basic TASEP, and investigate extensions to the model relevant to these biological transport processes. Applications to the translation of mRNA containing slowly translated codons will be discussed.