Polymer networks are an important class of materials that are ubiquitously found in natural, biological, and man-made systems. The complex mesoscale structure of these soft materials has made it difficult for researchers to fully explore their properties. In this dissertation, we introduce a coarse-grained computational model for permanently cross-linked polymer networks than can properly capture common properties of these materials. We use this model to study several practical problems involving dry and solvated networks. Specifically, we analyze the permeability and diffusivity of polymer networks under mechanical deformations, we examine the release of encapsulated solutes from microgel capsules during volume transitions, and we explore the complex tribological behavior of elastomers. Our simulations reveal that the network transport properties are defined by the network porosity and by the degree of network anisotropy due to mechanical deformations. In particular, the permeability of mechanically deformed networks can be predicted based on the alignment of network filaments that is characterized by a second order orientation tensor. Moreover, our numerical calculations demonstrate that responsive microcapsules can be effectively utilized for steady and pulsatile release of encapsulated solutes. We show that swollen gel capsules allow steady, diffusive release of nanoparticles and polymer chains, whereas gel deswelling causes burst-like discharge of solutes driven by an outward flow of the solvent initially enclosed within a shrinking capsule. We further demonstrate that this hydrodynamic release can be regulated by introducing rigid microscopic rods in the capsule interior. We also probe the effects of velocity, temperature, and normal load on the sliding of elastomers on smooth and corrugated substrates. Our friction simulations predict a bell-shaped curve for the dependence of the friction coefficient on the sliding velocity. Our simulations also illustrate that at low sliding velocities, the friction decreases with an increase in the temperature. Overall, our findings improve the current understanding of the behavior of polymer networks in equilibrium and non-equilibrium conditions, which has important implications for synthesizing new drug delivery agents, designing tissue engineering systems, and developing novel methods for controlling the friction of elastomers.