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Triangulations are a natural generalization of the concept of partitioning an interval into sub intervals. Triangulations form a huge subject in Mathematics. Triangulation of a planar disc is called by patches. The definition of patch first appeared in the literature in a paper by W. T. Tutte. A triangulation is said to be elliptic if it does not contain any point with degree greater than 6. This book provides the different types of elliptic triangulations for planar discs and spheres. Patches were constructed by different types of methods. The enlargement methods for constructing patches were also discussed in this book. The gluing methods of patches for constructing elliptic triangulations were also explained. It was shown that there were only 19 possible types of elliptic triangulations for spheres and determined the existence (as well as nonexistence) of all these types except for a small number of cases.