Unsteady aerodynamics of flapping wings and airfoils are in the center of research for quite some time in order to understand the flight of natural locomotion systems as well as the design of futuristic bio-mimetic devices. The present work examines the details of the unsteady flow-field past a flapping system, under different kinematic situations. Nonlinear dynamical behaviour and dynamical transitions of the unsteady wake are in the main focus. The initial part of the thesis explores the efficacy of a body non-conformal mesh method, a discrete forcing immersed boundary method (IBM), in capturing the transitional dynamics in the flow-field of a flapping foil in the low Reynolds number (Re) regime. The IBM based Navier-Stokes (N-S) solver was developed using the C++ language (parallelized with OpenMP technique) and extensively tested for a variety of flow problems involving stationary/oscillating bluff/streamlined bodies subjected to different flow conditions. In the later part of the thesis, interesting flow phenomena such as wake deflection, jet-switching and irregular wake patterns were identified and studied in detail. The associated nonlinear dynamical states, including chaos, and the transition (bifurcation) routes were also established using appropriate tools from the dynamical systems theory. The study also explores the role of fluctuating inflow and flexibility in controlling the qualitative nature of the wake, as well as their quantitative onsets. The change in the corresponding dynamical states and restoration of the periodic behaviour in the flow-field have also been investigated. The underlying flow physics are investigated through a qualitative study of the near-field interactions as well as various quantitative measures derived from the unsteady flow-field. A novel order-to-chaos map considering all the relevant kinematic parameters is proposed in order to differentiate the distinct dynamical wake regimes in a robust manner.