Many practical applications such as real-time control systems rely on reliable state estimates for their accurate functionality. The process of estimating a latent variable vector, called system state, using sensor measurements is known as state estimation. During recent years, the proliferation of smart sensing and computing devices connected to each other and on internet has enabled not only the realization of different innovative applications in various fields but also has changed the operations of many existing fields. At the same time, the availability of various different data streams in their raw forms is posing greater challenges to engineers/practitioners. As an example, in the context of state estimation of a cyber-physical system (CPS), the measurement and control data are often sensed using low-cost devices, processed in its raw form, and transmitted through unprotected communication links. This operational flow of the data is vulnerable to various different threats. These threats can be intentional e.g., cyberattacks, or unintentional such as sensor malfunctioning. In the presence of these threats, the correct functionality of system under consideration is not possible. In this thesis, we explore the process of robust state estimation using data which has been corrupted intentionally and unintentionally. We develop a class of algorithms which are based on classical observer principle but differ in terms of the norm of the error vector being used. We consider a more generic formulation based on L_p norm minimization and use gradient-based optimization for state estimation of non-linear dynamical system. In the first part, our focus was on testing the proposed family of algorithms on unintentional data corruptions such as sporadic sensor malfunctioning. In the second part, we have focused on intentional data corruptions such as cyberattacks for physical layer of CPS. For this purpose, we have modeled false data injection attacks and have analyzed the performance of the proposed algorithm in their presence.