The Back and Forth Nudging (BFN) algorithm is a prototype of a new class of data assimilation methods, although the standard nudging algorithm is known for a couple of decades. It consists in adding a feedback term in the model equations, measuring the difference between the observations and the corresponding space states. The idea is to apply the standard nudging algorithm to the backward (in time) nonlinear model in order to stabilize it. The BFN algorithm is an iterative sequence of forward and backward resolutions, all of them being performed with an additional nudging feedback term in the model equations. We also present the Diffusive Back and Forth Nudging (DBFN) algorithm, which is a natural extension of the BFN to some particular diffusive models. These nudging-based algorithms can be extended to more complex observers, with the aim of correcting non-observed variables, and improving the convergence of the algorithm and estimation of the model state, but also with the aim of identifying model parameters.