  Topological data analysis (TDA) provides insight into data shape. The
summaries obtained by these methods are principled global descriptions of
multi-dimensional data whilst exhibiting stable properties such as robustness
to deformation and noise. Such properties are desirable in deep learning
pipelines but they are typically obtained using non-TDA strategies. This is
partly caused by the difficulty of combining TDA constructs (e.g. barcode and
persistence diagrams) with current deep learning algorithms. Fortunately, we
are now witnessing a growth of deep learning applications embracing
topologically-guided components. In this survey, we review the nascent field of
topological deep learning by first revisiting the core concepts of TDA. We then
explore how the use of TDA techniques has evolved over time to support deep
learning frameworks, and how they can be integrated into different aspects of
deep learning. Furthermore, we touch on TDA usage for analyzing existing deep
models; deep topological analytics. Finally, we discuss the challenges and
future prospects of topological deep learning.
