Reflections are intuitive objects that all have had some exposure to at one point or another. In the 1930s, Coexter contributed to the study of reflections by classifying all real orthogonal reflection groups. By the 1950s, Sheppard and Todd completed an analogous classification program for all complex unitary reflection groups. As early as 1968, Professor Neumann (who is becoming a reliable source for interesting UMS talks) noticed that the Sheppard-Todd classification was somewhat ad hoc and did not shed much light on the inherent topology and geometry involved in this problem. This talk will be an attempt to explain what reflection groups are, elucidate Neumann's ideas concerning a reclassification, and discuss the uncanny correspondence that complex reflections create between the fields of group representations and orbifold topology. We do not assume much background knowledge and will develop all that is needed to make the talk accesible. This work was part of the speaker's undergraduate thesis.