Data assimilation (DA) is the process of optimally combining observations with the predictions of numerical models to make the best possible estimate of the time-varying state of the phenomenon under study. In particular, DA forms a basis for the forecast of the future and re analysis of the past. In the last 20 years, DA has gained center stage in many computational disciplines at both universities and research centers starting with geoscience applications. DA is a subject that requires a balanced understanding of statistics and applied mathematics as well as the relevant geophysical systems. This course introduces the concepts of data assimilation derived in the context of the statistical estimation theory and the deterministic inverse theory. The course covers a variety of assimilation methods for numerical weather prediction, ocean forecasting, reservoir history matching, 4D seismic inversion, and hydrology assimilation. These include, but not limited to, optimal interpolation and three (3) dimensional variational (3D VAR) methods, Kalman filtering, smoothing and fourdimensional variational (4D VAR) methods, low rank Kalman filtering, ensemble Kalman filtering and ensemble square-root filters. Advanced topics based on the fully nonlinear Bayesian estimation theory, such as the particle filter and the Gaussian Mixture filters, and the state of art data assimilation systems will also be discussed.