This thesis concerns two-point sets, which are subsets of the real plane which intersect every line in exactly two points. The existence of two-point sets was first shown in 1914 by Mazukiewicz, and since this time, the properties of these objects have been of great intrigue to mathematicians working in both topology and set theory. Arguably, the most famous problem about two-point sets is concerned with their so-called "descriptive complexity"; it remains open, and it appears to be deep. An informal interpretation of the problem, which traces back at least to Erdos, is: The term "two-point" set can be defined in a way that it is easily understood by someone with only a limited amount of mathemat- ical training. Even so, how hard is it to construct a two-point set? Can one give an effective algorithm which describes precisely how to do so? More formally, Erdos wanted to know if there exists a two-point set which is a Borel subset of the plane. An essential tool in showing the existence of a two-point set is the Axiom of Choice, an axiom which is taken to be one of the basic truths of mathematics.