This book presents the development and application of some topological methods in the analysis of data coming from 3D dynamical systems (or related objects). The aim is to emphasize the scope and limitations of the methods, what they provide and what they do not provide. Braid theory, the topology of surface homeomorphisms, data analysis and the reconstruction of phase-space dynamics are thoroughly addressed.Contents:A Crisis in the Experimental Method ArchetypeOrbit Organization in R2 x S1Braids as Indicators of Phase-Space DynamicsBraids and the Poincaré SectionReconstruction of Phase-Space Dynamics — Basic CourseReconstruction of Phase-Space Dynamics — Advanced CourseReadership: Researchers in geometry and topology and mathematical modeling.