Data Assimilation aims to optimally combine dynamics models of physical systems with available measurements to estimate a state or parameter of the system. Many systems or observations naturally live on smooth manifolds, while most sequential classical data assimilation algorithms, including Kalman and its extension to non-linear dynamics, are defined in Euclidean spaces. Typically, one can use an estimate-then-project method to address this problem. However, this is purely empirical, and an embedding in the Euclidean space needs to be accessed. Another strategy is implementing a classical filter algorithm in local coordinates when possible. However, this approach ignores the geometric structure of state and data spaces. In this talk, we propose to develop a framework for computing optimal filters in general manifolds and deterministic data assimilation setting. Considering a continuous-time model with discrete-time observations of a real trajectory, we first derive a formulation of the Mortensen filter in discrete-time in the context of manifold spaces for state and/or data. Optimal estimators are derived from the solution of a Hamilton-Jacobi-Bellman equation in the state space. We then derive a consistent continuous-time version and a spatial discretization. In the second part, we reduce the cost of the resulting algorithm by using a quadratic approximation of the value function solution of the Hamilton-Jacobi-Bellman equation, retrieving and justifying a version of the Extended Kalman filter adapted to the manifold context. We also investigate a tensor-based approximation, which can be interpreted as a more general class of approximation. We provide numerical state and parameter estimation results using these methods, in particular with the challenging example of a chaotic system: the planar double pendulum with observations on the position. Finally, we show how our approach can be adapted to deal with shape data defined in adapted space manifolds. We provide illustrations by the assimilation of models taken from environmental sciences - in particular wildfire models.