Explore the intricacies of statistical manifolds and their tangent bundles in this illuminating 24-minute conference talk presented at GSI. Delve into the mathematical foundations and geometric properties of these complex structures, gaining insights into their applications in statistical analysis and information geometry. Examine the relationship between statistical manifolds and their associated tangent spaces, uncovering the fundamental principles that govern their behavior and interactions. Enhance your understanding of differential geometry in the context of statistical theory, and discover how these concepts contribute to advancements in machine learning, data analysis, and statistical inference.