Normal DNA replication is blocked by DNA damage in the template strand. Translesion synthesis is a major pathway for overcoming these replication blocks. In this process, multiple non-classical DNA polymerases form a complex at the stalled replication fork called the mutasome. This complex is structurally organized by the replication accessory factor PCNA and the non-classical DNA polymerase Rev1. One of the non-classical DNA polymerases within the mutasome then catalyzes replication through the damage. Each non-classical DNA polymerase has one or more cognate lesions, which the enzyme bypasses with high accuracy and efficiency. Thus, the accuracy and efficiency of translesion synthesis depends on which non-classical DNA polymerase within the mutasome is chosen to bypass the damage. In this thesis, I discuss how the most appropriate polymerase is chosen. In so doing, I examine the components of the mutasome; the structural motifs that mediate the protein interactions in the mutasome; the methods used to study translesion synthesis; the definition of a cognate lesion; the intrinsically disordered regions that tether the polymerases to PCNA and to one another; the multiple architectures that the mutasome can adopt, such as PCNA tool belts and Rev1 bridges; and the kinetic selection model in which the most appropriate polymerase is chosen via a competition among the multiple polymerases within the mutasome. Taken together, this thesis provides and inclusive review of the current state of what is known about translesion synthesis with conclusions at its end suggesting what major questions remain and ideas of how to answer them.