In this thesis we consider degree sequences of triangulations of point sets in convex position in the plane. This is equivalent to degree sequences of maximal outerplanar graphs. Utilizing basic properties as well as new relations for triangulations we develop sufficient and necessary conditions for special cases of nonnegative integer sequences to be valid degree sequences of a triangulation of a convex point set. This includes the more general cases where we have more 'big nodes' (nodes with degree greater than 2) than nodes with degree 0 (ears). Also for this case we present sufficient and necessary conditions for a sequence to be a degree sequence. Additionally, we present algorithmic constructions of canonical triangulations for all discussed cases.