There has been a great deal of attention paid to "Big Data" over the last few years. However, often as not, the problem with the analysis of data is not as much the size as the complexity of the data. Even very small data sets can exhibit substantial complexity. There is therefore a need for methods for representing complex data sets, beyond the usual linear or even polynomial models. The mathematical notion of shape, encoded in a metric, provides a very useful way to represent complex data sets. On the other hand, Topology is the mathematical sub discipline which concerns itself with studying shape, in all dimensions. In recent years, methods from topology have been adapted to the study of data sets, i.e. finite metric spaces. In this talk, we will discuss what has been
done in this direction and what the future might hold, with numerous examples.