The book "Geometric Science of Information" presents the refereed proceedings of the second International Conference on the Geometric Science of Information, which took place in October 2015 in Palaiseau, France. It includes 80 full papers that were carefully selected from 110 submitted works. The contributions are divided into thematic sessions covering various aspects of the geometric science of information, including dimensionality reduction on Riemannian manifolds, optimal transport methods, applications in image processing and statistics, as well as the geometry of time series. This work provides a comprehensive overview of current research findings and developments in this interdisciplinary field that connects mathematics, computer science, and statistics.