Abstract: Topological data analysis (TDA), a new approach to handling high-dimensional data, gained a lot of attention lately. TDA focuses on qualitative rather than quantitative information supported by the data. A central concept in TDA is persistent homology, a topological invariant capturing structural changes in continuous objects reflected by the discrete point clouds. In particular, persistent homology allows one to determine the thresholds where new topological features emerge and where they are later destroyed, providing a so-called birth-death decomposition. Birth-death decompositions are often represented as collections of planar points called persistence diagrams, and they have been extensively used in a variety of applications. Often, continuous objects associated with a point cloud are sublevel sets of a function on a manifold. In such a case homological duality results may lead to nice relations between persistence diagrams in different homological dimensions, which doesn't only provide more insight into the theory of persistent homology, but can also be used in practice to reduce computational time. In this talk, I will review some of the existing duality results in persistent homology and show that by focusing not on persistence diagrams but on persistence modules, which are algebraic objects representing birth-death decompositions, one can obtain significantly more general duality results.