To propose e cient and better designs for small swimming and ying unmanned vehicles, understanding of the unsteady mechanisms to generate lift and thrust forces at low Reynolds numbers is of key importance. Fluid owing over these vehicles interact nonlinearly with the structure and carries great complexities. Recently, due to interest in biomimicking ying (micro-air vehicles) and swimming robots (underwater vehicles), industry has shown keen interest in production of these vehicles. To design e ective control of these vehicles, thorough understanding of its unsteady aerodynamics and underlying phenomena is required. In this study, we focus upon coupling the numerical simulations with the tools of nonlinear dynamics. We decompose this whole study into two parts; aerodynamics and hydrodynamics. In therst part, we investigate the bifurcations occurring in the ows over oscillating airfoils at low Reynolds numbers. Investigation of mechanism responsible for the generation of unsteady forces pose challenges due to wide spectrum of parameters that are involved in its dynamics. Both experimental and currently available numerical techniques require costly resources in terms of time and money. Considering this fact, we also develop nonlinear reduced-order models for unsteady aerodynamic forces produced by plunging, pitching, and apping airfoils. Observing similarity in the character of unsteady forces generated by pitching, and plunging airfoils, we propose an equivalence criteria to obtain the aerodynamic forces of same magnitude or order. We also demonstrate that de ection of the wake for large Strouhal numbers is a result of strong quadratic nonlinearity. With the lessons learnt from the nonlinear analysis/interaction of apping airfoils, we investigate the hydrodynamics ofsh swimming in the second part of this dissertation. We consider a singlesh and twosh in tandem performing traveling-wave like motion, known as undulation. In case of tandem con guration, we numerically simulate the ow while both sh undulate asynchronously. We quantify the drafting and inverse-drafting e ects using time-averaged drag coe cients. We also explain physical mechanisms which are responsible for hydrodynamic advantage/disadvantage to upstream and downstreamsh. To further enhance our understanding related to the instability mechanisms in the wakes of undulating bodies, we compute the symmetry/asymmetry of parent and combined modes. We apply the symmetry principles, already established for drag-producing wakes of blubodies, to the thrust-producing wakes of undulatingsh. We conclude that thrust producing wakes also follow the same symmetry principles. This research addresses the coupling of techniques/tools of nonlinear mechanics with computational uid dynamics to explore important features of complex ows around oscillating and undualating bodies.