Since the 1970s, applications of Bayesian time series models and forecasting methods have represented major success stories for our discipline. Dynamic modelling is a very broad field, so this ISBA Lecture on Bayesian Foundations will rather selectively note key concepts and some core model contexts, leavened with extracts froma few time series analysis and forecasting examples from various application fields. The Lecture with then link into and briefly discuss a range of recent developments in exciting and challenging areas of Bayesian time series analysis.