  We provide a short introduction to the field of topological data analysis and
discuss its possible relevance for the study of complex systems. Topological
data analysis provides a set of tools to characterise the shape of data, in
terms of the presence of holes or cavities between the points. The methods,
based on notion of simplicial complexes, generalise standard network tools by
naturally allowing for many-body interactions and providing results robust
under continuous deformations of the data. We present strengths and weaknesses
of current methods, as well as a range of empirical studies relevant to the
field of complex systems, before identifying future methodological challenges
to help understand the emergence of collective phenomena.
