There are two main aims of this thesis. The first is to further develop and demonstrate applications of the combinatorial interpretation of continued fractions introduced in [Benjamin and Quinn, 2003]. The second is to investigate the theory of negative continued fractions, a relatively unresearched topic. That is, discuss the ways in which they are similar to and different from the regular class, describe how to convert between the two forms, and show that the central theorems concerning regular continued fractions also apply to the negative ones.