Category theory provides a powerful framework for understanding abstract structures in mathematics, enabling us to identify structural similarities across seemingly unrelated areas by working at a high level of abstraction. Infinity-categories, an advanced extension of category theory, facilitate the study of mathematical structures with richer homotopical data. In this context, the morphisms between objects form not just a set, but a more intricate object such as a topological space or a chain complex. In this seminar, we will explore the motivations behind this shift to infinity-categories and examine their applications in areas like algebraic topology, algebraic geometry and homotopy theory.