The effects of viscosity and elasticity stratification on interfacial instabilities of symmetric and asymmetric three-layer superposed pressure driven channel flows have been investigated theoretically using linear stability analysis. The results indicate that the presence of an additional interface has a significant effect on the stability of the flow. Specifically, it has been shown that resonant instabilities as well as finite wavenumber interfacial instabilities could occur in this class of flows. Moreover, it has been demonstrated that a thin layer of less viscous fluid adjacent to the solid wall can stabilize longwave disturbances in both symmetric and asymmetric superposed Newtonian flows. In addition, in asymmetric superposed Newtonian flows it has been shown that when the jump in the shear rate across both the interface is positive then both the interfacial modes are unstable. In case of purely elastic instabilities, longwave disturbances can be stabilized if the more elastic fluid occupies the majority of the channel, irrespective of its position. Moreover in both purely elastic and viscous instabilities the dominant mode of the instability has been shown to have wavenumbers of O(1). It has also been demonstrated that stability of viscoelastic flows can not be deduced based on the guidelines developed for purely elastic and viscous instabilities (i.e., the effects are non-additive). Furthermore, nonlinear constitutive equations which accurately depict the steady as well as transient viscoelastic properties of typical polymeric melts and solutions with various degrees of flexibility and accuracy have been used to assess the constitutive complexity required to accurately describe the stability characteristics of this class of flows by comparing the results of the stability analysis with the experimental results of Khomami and Ranjabaran [9]. It is shown that the multimode Giesekus model, which can accurately describe the steady as well transient behavior of the polymeric test fluids used in the experiments, can quantitatively describe the interfacial instability phenomenon in terms of the neutral stability contour as well as the growth/decay rate behavior when the effect of interfacial tension is taken into account. A rigorous energy analysis based on a disturbance-energy equation for viscoelastic flows has also been performed to investigate the mechanism of the purely viscous and purely elastic interfacial instabilities in pressure driven channel flows. The mechanisms of purely viscous and purely elastic instabilities of the three-layer flows are found to be identical to those of the two-layer flows. Namely, the mechanism of shortwave purely viscous instability is found to be due to the viscosity mismatch and the subsequent perturbation vorticity mismatch at the interface (i.e., interfacial friction), whereas the mechanism of the longwave purely viscous instability is found to be due to the bulk Reynolds stresses. The mechanism of purely elastic instability is found to be due to the coupling between the perturbation velocity and the jump in normal stresses across the interface at longwaves as well as shortwaves. Finally, the possibility of non-normal interactions between the two interfacial modes or interfacial and bulk modes has been demonstrated and it has been shown that these interactions should be carefully considered when the disturbance growth/decay rates are experimentally determined.