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Every moment of our daily life belongs to the new era of" Big Data". We continuously produce, at an unpredictable rate, a huge amount of heterogeneous and distributed data. The classical techniques developed for knowledge discovery seem to be unsuitable for extracting information hidden in these volumes of data. Therefore, there is the need to design new computational techniques. In this paper we focus on a set of algorithms inspired by algebraic topology that are known as Topological Data Analysis (TDA). We brie y introduce the principal techniques for building topological spaces from data and how these can be studied by persistent homology. Several case studies, collected within the TOPDRIM (Topology driven methods for complex systems) FP7-FET project, are used to illustrate the applicability of these techniques on different data sources and domains.