Birational non-rigidity of codimension 4 Fano 3-folds

In this thesis we prove the birational non-rigidity of Picard rank 1 Fano 3-folds in codimension 4 having Fano index 1. This is done by explicitly constructing Sarkisov links for these varieties to other Mori fibre spaces. We also consider those Fano 3-folds in codimension 4 and Fano index 1 having Picard rank 2, and we identify a Mori fibre space in its birational equivalence class. In a final short chapter, we begin this program for Fano 3-folds in codimension 4 having Fano index 2 by demonstrating a construction of them as quotients of index 1 Fano 3-folds.