Multilayer composite structures are used in a wide variety of applications, from sporting goods to aerospace engineering and in robotic arms and floor joists. A common design for a multilayer composite structure consists of n = 2m + 1 layers, in which m+1 stiff layers are bound together by m shear-deformable layers. It has been known for 50 years that the shear motion in the compliant layers is responsible for most of the damping of flexural vibrations. We consider multilayer beam and plate models in which linear viscous shear damping is included in the shear-deformable layers. We formulate the equations of motion for such a structure as a partial differential equation (PDE) semigroup problem, and we use the theory of PDE semigroups to prove stability results for damped multilayer beams and plates. In particular, we show that the semigroups associated multilayer beam and plate models of Mead and Markus are both analytic and exponentially stable, and we show that the semigroup associated with the multilayer beam of Rao and Nakra is exponentially stable under certain conditions. In addition, we consider two optimal damping problems for the multilayer Mead-Markus beam: i.) choosing damping parameters in the shear-deformable layers to achieve the optimal angle of analyticity, and ii.) choosing damping parameters in the shear-deformable layers to achieve the optimal energy decay rate.