Two subjects related to the non-equilibrium dynamics of polymers or biological filaments adsorbed to two-dimensional substrates will be discussed. The first part is dedicated to thermally activated dynamics of polymers on structured substrates in the presence or absence of a driving force. The structured substrate is represented by double-well or periodic potentials. Both homogeneous and point driving forces are considered. Point-like driving forces can be realized in single molecule manipulation by atomic force microscopy tips. Uniform driving forces can be generated by hydrodynamic flow or by electric fields for charged polymers. The second part is dedicated to collective filament motion in motility assays for motor proteins, where filaments glide over a motor-coated substrate. The model for the simulation of the filament dynamics contains interactive deformable filaments that move under the influence of forces from molecular motors and thermal noise. Motor tails are attached to the substrate and modeled as flexible polymers (entropic springs), motor heads perform a directed walk with a given force-velocity relation. I will consider the collective filament dynamics and pattern formation as a function of the motor and filament density, the force-velocity characteristics, the detachment rate of motor proteins and the filament interaction. In particular, the formation and statistics of filament patterns such as nematic ordering due to motor activity or clusters due to blocking effects are investigated.