The surface oscillations in a deformable wall are known to induce an instability in the adjacent flow even in the absence of inertia. This instability, if understood properly, can be exploited to generate a well-mixed flow pattern with improved transport coefficients in microfluidic systems, wherein the benefits of inertial instabilities can not be realised. In order to utilise the wall deformability in micro-devices as well as other biotechnological applications, the quantitative knowledge of the critical parameter for the on-set of instability and the nature of bifurcation in the region of transition point are essential. With this objective, a major portion of this thesis deals with the stability analysis of flow past a flexible surface. For Newtonian flow over a deformable solid medium, the analyses of hydrodynamic stability in two flow regimes are presented: the viscous mode instability in the limit of zero Reynolds number, and the wall mode instability in the limit of high Reynolds number. The flexible solid in both analyses is described as a neo-Hookean solid continuum of finite thickness. The previous work on viscous instability using the same solid model ignored the viscous dissipation in the solid. In the present study, a purely elastic neo-Hookean model is augmented to incorporate the viscous stresses accounting for the dissipative mechanism in an aqueous gel-like solid medium. The linear stability analysis for this neo-Hookean viscoelastic solid shows a dramatic influence of solid viscosity on the stability behaviour. The important parameter here is where ηr is the solid viscosity relative to the fluid viscosity and H is the solid-to-fluid thickness ratio. While the effect solid viscosity is stabilizing for a further increase in viscosity in the regime reduces the critical shear rate for transition, indicating a destabilizing influence of solid viscosity. The weakly nonlinear analysis indicates that the bifurcation is subcritical for most values of H when ηr =0. However, for non-zero solid viscosity, the analysis reveals a range of ηr for which the nature of bifurcation is supercritical. The results are in contrast to the behaviour for the Hookean (linear) elastic solid, for which the effect of solid viscosity is always stabilising and the bifurcation is subcritical for all values of H and ηr. For the wall mode instability, critical parameters for the linear and the neo-Hookean elastic solid are found to be very close. The weakly nonlinear analysis of the wall mode instability shows that the instability is driven to a supercritically stable branch, indicating the possibility of a stable complex flow pattern which is ) correction to the base flow. The amplitude of the supercritically bifurcated equilibrium state, A1e, is derived in the vicinity of the critical point, and its scaling with the flow Reynolds number is obtained. The nonlinear analysis is also carried out using the asymptotic analysis in small parameter Re−1/3. The asymptotic results are found to be in good agreement with the numerical solutions for For a polymeric flow over a deformable solid medium, the viscous instability is analysed by extending the viscous mode for the Newtonian fluid to the fluid with finite elasticity. The viscoelastic fluid is described by an Oldroyd-B model which introduces two additional parameters: the Weissenberg number, W , and β, the ratio of solvent-to-solution viscosity. The polymer viscosity parameter β is an indirect measure of polymer concentration with the extreme cases of β =1 representing the Newtonian fluid and β =0the upper convected Maxwell fluid. The analysis considers both the linearly elastic and the neo-Hookean models to describe the deformable solid. The analysis reveals the presence of two classes of modes: the finite wavelength modes and the shortwave modes. The behaviour of the finite wavelength modes is similar for both the models of solid medium. The effect of increasing fluid Weissenberg number and also increasing polymer concentration (achieved by reducing β below 1) on the finite wavelength instability is stabilising. The viscous instability ceases to exist for W larger than a certain maximum value Wmax. The behaviour of the shortwave mode is remarkably different for both the models of solid. Using the shortwave asymptotic, the differences are elucidated and it is shown that the shortwave instabilities in both the models are qualitatively different modes. For a linear elastic solid model, the shortwave mode is attributed to the normal-stresses in polymeric fluid with high Weissenberg number. This mode does not exist for the Newtonian flow and is a downstream travelling disturbance wave. On the other hand, the shortwave mode for the neo-Hookean model is attributed to the normal-stress difference in the elastic solid. Hence, this mode does exist for the Newtonian fluid and is an upstream travelling disturbance wave. The role of polymer concentration in the criticality of finite wavelength and shortwave modes is examined for a wide range of Weissenberg number. The results are condensed in a map showing the stability boundaries in parametric space covering β, W and H. The weakly nonlinear analysis reveals that the bifurcation of linear instability is subcritical when there is no dissipation in the solid. The nature of bifurcation, however, changes to supercritical when the viscous effects in the solid are taken into account. The final problem of this thesis deals with the flow past a rigid surface. Here, the stability of base profile in a plane Couette flow of dilute polymeric fluid is studied at moderate Reynolds number. Three variants of Oldroyd-B model have been analysed, viz. the classical Oldroyd-B model, the diffusive Oldroyd-B model, and the non-homogeneous Oldroyd-B model. The Newtonian wall modes are modified marginally for the polymeric fluid described by the classical Oldroyd-B model. The Oldroyd-B model with artificial diffusivity introduces the additional ‘diffusive modes’ which scale with P´eclet number. The diffusive modes become the slowest decaying modes, in comparison to the wall modes, for large wavenumber disturbances. For these two models, the polymeric flow is linearly stable. Using the equilibrium flow method, wherein the nonlinear flow is assumed to be at the transition point, the finite amplitude disturbances are analysed, and the threshold energy necessary for subcritical transition is estimated. The third variant of Oldroyd-B model accounts for non-homogeneous polymer concentration coupled with the stress field. This model exhibits an instability in the linear analysis. The ‘concentration mode’ becomes unstable when the fluid Weissenberg number exceeds a certain transition value. This instability is driven by the stress-induced fluctuations in polymer number density.