In May of 2000 at a conference in Paris, one hundred years after David Hilbert famously presented a list of what he saw as the 23 most significant unsolved problems in mathematics, the Clay Mathematics Institute made a historic announcement. The Institute would award a prize of one million dollars for the solution to each of seven unsolved problems in mathematics. These problems, known as the Millennium Problems, are among the most difficult and important problems in mathematics. We will focus our attention on the only Millennium Problem regarding computers, the P vs. NP Problem. Most, if not all, mathematicians and computer scientists consider this to be the most important question in computer science. Roughly speaking, this problem asks, "If solutions to a problem can be verified quickly, then can they also be found quickly?" We will make this question more precise as we carefully attempt to understand this significant problem and its consequences in mathematics and computer science. http://www.brynmawr.edu/math/colloquium.html