Data assimilation is a group of mathematical methods aimed at adjusting a model of interest with available data. The methods covered in this thesis can be divided into two categories: variational and sequential methods. Variational methods minimize the cost function that measures the accuracy of the prediction compared to the model and observations, in a specified time interval. Two most commonly used algorithms are 4D-Var algorithm and 3D-Var as its special case. Incremental 4D-Var algorithm minimizes a sequence of linear convex cost functions, instead of a nonlinear one as in 4D-Var. Sequential methods for data assimilation make use of observations as soon as they become available. One such sequential algorithm is the Kalman filter, in which the operators used in the model definition are assumed to be linear, and errors are assumed to be normally distributed. These limitations are dropped in Extended Kalman filter algorithm which uses linearizations of those operators for some steps. Furthermore, the matrices used in the method can be replaced by their low-rank approximations, for better numerical performance. That leads to a completely new set of methods. In this thesis only a small part of a rather large set of algorithms for data assimilation is introduced and illustrated on practical examples.