We are rarely able to fully and directly observe many phenomena which are crucial to our daily lives. However, it is often the case that certain partial information on the phenomena of interest is available together with a mathematical model of it. The general question that one is interested in this case is what inferences can be done on the phenomena based on the partial data and the prior assumptions. In
general, the evolution of the real life phenomena is a (partially observed) dynamical system. This naturally leads us to the area of stochastic filtering, which is defined as the estimation of dynamical systems whose trajectory is modelled by a stochastic process called the signal, given the information accumulated from its partial observation. Various applications of the problem include the control of engineering systems, data assimilation in meteorology, volatility estimation in financial markets, computer vision and vehicle tracking. A massive scientific and computational effort is dedicated to the development of various tools for approximating the solution of the filtering problem. In this thesis, we cover several topics related to the numerical approximation of the solution of the filtering problem. The first topic is related to Sequential Monte Carlo methods, or particle filters. Here we analyse the bias resulting from such approximations. We then study the filtering problem associated to the Cox-Ingersoll-Ross model. This model is ubiquitous in many financial applications as it describes the evolution of interest rates. The third topic is related to the projection filter. We first provide a local convergence result and then propose a way to generalize it to a global result. The thesis also contains results related to the stability of the filtering solution and the Navier-Stokes equation.