The topic of functional time series has received some attention recently. This is timely as many applications involving space-time data can benefit from the functional-data perspective. In this talk, I will start off with the Argo data, which have fascinating features and are highly relevant for climate research. I will then turn to some extensions of stationarity in the context of functional data. The first is to adapt the notion of intrinsic random functions in spatial statistics, due to Matheron, to functional data. Such processes are stationary after suitable differencing, where the resulting stationary covariance is referred to as generalized covariance. A Bochner-type representation of the generalized covariance as well as preliminary results on inference will be presented. The second extension considers intrinsic stationarity in a local sense, viewed from the perspective of so-called tangent processes. Motivations of this work can be found from studying the multifractional Brownian motion.