The problem of a thin elastomeric layer confined between two stiff adherends arises in numerous applications such as microelectronics, bio-inspired adhesion and the manufacture of soft biomedical products. A common requirement is that the debonding of the elastomeric layer from the adherends be controlled to avoid undesirable failure modes. This level of control may necessitate understanding the collective role of the interfacial adhesion, material properties, part geometries, and loading conditions on the debonding. Analytical and numerical approaches using the finite element method and a cohesive zone model (CZM) for the interfacial debonding are used in this dissertation to delineate the role of the afore-mentioned parameters on the initiation and propagation of debonding for both rigid and non-rigid adherends. Extensively studied in the dissertation is the debonding of a semi-infinite relatively stiffer adherend from an elastomer layer with its other surface firmly bonded to a rigid base. The adherend is pulled upwards by applying normal displacements either on its entire unbonded surface or on the edge of its part overhanging from the elastomer layer. The adherend and the elastomeric layer materials are assumed to be linear elastic, homogeneous and isotropic and the elastomer is assumed to be incompressible. Viscoelasticity of the elastomer is considered in the first part of the work. Plane strain deformations of the system with a bilinear traction-separation (TS) relation in the CZM are analyzed. Two non-dimensional numbers, one related to the layer confinement and the other to the interfacial TS parameters, are found to determine if debonding initiates at interior points in addition to at corner points on the adherend/elastomer interface, and if adhesion-induced instability is exhibited. This work is extended to axisymmetric problems in which debonding can take place at both interfaces. Motivated by an industrial demolding problem, numerical experiments are conducted to derive insights into preferential debonding at one of the two interfaces, including for curved adherends. Results reported herein should help engineers design an elastomer layer sandwiched between two adherends for achieving desired failure characteristics. / Ph. D.