<p>Topological data analysis is a powerful tool for describing topological signatures in real world data. An important challenge in topological data analysis is the task of matching significant topological signals across distinct systems. Optimal transportation provides a geometric and probabilistic approach to formalising notions of distances and matchings between measures and structured objects more generally. Building upon recent advances in the domains of persistent homology and optimal transport for hypergraphs, we develop an approach for finding geometrically and topologically informed matchings between point cloud datasets and their topological features. We define measure topological networks, which integrate both geometric and topological information about a system, and introduce a distance on the space of these objects. Our distance is defined in terms of a Topological Optimal Transport problem which seeks to transport mass that preserves geometric as well as topological relations, in the sense of minimising the induced geometric and topological distortions. We study the metric properties of this distance and show that it induces a geodesic metric space of non-negative curvature. We demonstrate our approach in a number of numerical experiments.</p>