Euler integration is an integration theory with the Euler characteristic acting as the measure, and similar to classical analysis, it comes equipped with a collection of integral transforms. In this thesis, we focus on two such integral transforms: the persistent homology transform and the Fourier-Sato transform. We prove the invertibility of the former using the technique of Radon transform, and show the connection of the latter to Euler convolution and inner product. We also provide a new way to interpret the Euler integral through a generalization of combinatorial species, which also extends to magnitude homology and configuration spaces
