Topological Data Analysis (TDA) is the collection of mathematical tools that capture the structure of shapes in data. Despite computational topology and computational geometry, the utilization of TDA in time series and signal processing is relatively new. In some recent contributions, TDA has been utilized as an alternative to the conventional signal processing methods. Specifically, TDA is been considered to deal with noisy signals and time series. In these applications, TDA is used to find the shapes in data as the main properties, while the other properties are assumed much less informative. In this paper, we will review recent developments and contributions where topological data analysis especially persistent homology has been applied to time series analysis, dynamical systems and signal processing. We will cover problem statements such as stability determination, risk analysis, systems behaviour, and predicting critical transitions in financial markets.
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