Data assimilation tries to predict the most likely state of a dynamical system by combining information from observations and prior models, often represented by a discretized partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches. In particular, three-dimensional and four-dimensional variational data assimilation (3D-Var and 4D-Var), and, if time allows, the Kalman filter. The data assimilation problem usually results in a very large, yet structured, nonlinear optimization problem. The dimension of the latter represents a quite challenging aspect of the entire solution procedure. Indeed, the inclusion of both the time and space dimensions leads to extremely large optimazion problems which need to be carefully handled by designing smart numerical schemes able to fully exploit the structure of the problem at hand. Therefore, the second part of this talk aims to review advances and challenges, in particular from the numerical linear algebra perspective, within the various data assimilation approaches mentioned above.