Classical vector valued paths are widespread across pure and applied mathematics: from stochastic processes in probability to time series data in machine learning. Parallel transport of such paths in principal G-bundles have provided an effective method to characterise such paths. In this talk, we provide a brief overview of these results and their applications. We will then discuss recent work on extending this framework to characterizing random and possibly nonsmooth surfaces using surface holonomy. This is based on joint work with Harald Oberhauser.