Technological advances are providing researchers with larger and larger amounts of data, creating a rich landscape for data scientists to explore and search for meaningful patterns. Novel biomedical imaging techniques over the next 5 to 10 years, for instance, will develop their resolution to the point at which single subject scans might provide terabytes of data. While traditional data analytics cannot handle these large volumes of data, various Big Data statistical methods should be investigated and developed. This is clearly a great challenge for all stages of data management, but most importantly for the analysis and interpretation. In this chapter, we discusse a very recent paradigm and a set of techniques rooted in algebraic topology, which are able to scale and provide useful information within the context of abundant but often low-quality data sets. These techniques, collectively referred to as Topological Data Analysis, can extract meaningful information without the need to rely on specific a priori models or hypotheses. The potential computational and interpretative pitfalls as well as the potential benefits of using a robust-by-design high-order relational data description are also highlighted. Among these, we focus on persistent homology, its combination with machine learning, as well as topological simplification, together with their applications in biomedical signal processing.