Geometricians study objects and their symmetries, including the rotation of spheres. Lie groupoids, a powerful mathematical framework, emerge as a sophisticated tool for tackling these geometric problems by encapsulating both the objects and their symmetries in a unified language and preserving all the inherent algebraic and geometric structures. This enables us to examine seemingly disparate problems from a common perspective, revealing underlying connections and patterns. This information can be further refined using a concept known as the Lie algebroid, which serves as the “linear approximation” or “derivative” of the groupoid. Despite their seemingly simpler nature, Lie algebroids contain a wealth of information. he theory of Lie groupoids and Lie algebroids holds immense potential to amalgamate tools from geometry and algebra, yielding elegant and powerful results. Although this theory remains in its early stages of development, my project seeks to tackle fundamental questions at its core, with the potential to uncover applications in other realms of geometry and algebra.