The theory of rough paths, conceived in the 1990s, has recently evolved significantly beyond its origins in controlled and stochastic differential equations, witnessing a remarkable surge in applications within the realms of data science and machine learning. The central concept behind these advances is the signature transform, a map that captures a multidimensional data stream by sending it to the sequence of its iterated integrals. Intended as an easily accessible invitation to the field, this talk offers a natural functional-analytical perspective on the signature transform and, going beyond theoretical abstraction, gives a brief outlook on how related ideas from rough path theory can be applied to some contemporary challenges within stochastic analysis and statistical machine learning.