In this book the authors describe the principles and methods behind probabilistic forecasting and Bayesian data assimilation‎.‎ Instead of focusing on particular application areas‎,‎ the authors adopt a general dynamical systems approach‎,‎ with a profusion of low‎-dimensional‎,‎ discrete‎-time numerical examples designed to build intuition about the subject‎.‎ Part I explains the mathematical framework of ensemble‎-based probabilistic forecasting and uncertainty quantification‎.‎ Part II is devoted to Bayesian filtering algorithms‎,‎ from classical data assimilation algorithms such as the Kalman filter‎,‎ variational techniques‎,‎ and sequential Monte Carlo methods‎,‎ through to more recent developments such as the ensemble Kalman filter and ensemble transform filters‎.‎ The McKean approach to sequential filtering in combination with coupling of measures serves as a unifying mathematical framework throughout Part II‎.‎ Assuming only some basic familiarity with probability‎,‎ this book is an ideal introduction for graduate students in applied mathematics‎,‎ computer science