Viscoelasticity is a phenomenon of time-dependent strain and/or stress in elastic solids. Various contact interfaces with anthropomorphic end-effectors and polymeric solids found in robots and manipulators are intrinsically viscoelastic. It is therefore important to model such behavior and to study the effects of such time-dependent strain and stress on the stability and sustainability of grasping and manipulation. Both theoretical modeling and experimental study are presented in this dissertation. In theoretical modeling, a new nonlinear latency model is proposed for the application of contact interface involving viscoelasticity in robotics. Latency model can describe well various features of viscoelastic materials, such as stress relaxation, creep, and strain stiffening. The theoretical modeling was supported by experiments and computational simulation. Experiments were conducted by applying displacement-based control to study the stress relaxation and force-based control to explore the creep phenomenon, respectively, in order to validate the proposed theory. The experimental results of viscoelastic responses were observed, and found to match well with the proposed model as well as simulation results.