Explore the intricacies of moduli problems and abelianization in this Rothschild Public Lecture delivered by Professor Michael Thaddeus from Columbia University. Delve into the world of algebraic and differential geometry, examining how moduli spaces parametrize objects or structures and shed light on the original space's geometry. Discover the proposed abelianization process, which substitutes the action of an abelian group for more general group actions. Gain insights into two classical moduli problems: finitely many points in a projective space and vector bundles on a smooth curve. Presented at the Isaac Newton Institute for Mathematical Sciences, this one-hour lecture offers a deep dive into advanced mathematical concepts and their applications in geometry.