Random walks of molecular motors, which bind to and unbind from cytoskeletal filaments, are studied theoretically. The bound and unbound motors undergo directed and nondirected motion, respectively. Motors in open compartments exhibit anomalous drift velocities. Motors in closed compartments generate stationary nonequilibrium states with spatially varying densities of the motor concentrations and currents. "Traffic jams" on the filaments lead to a maximum of the motor current at an optimal motor concentration. Quantitative estimates based on experimental data for bound motors indicate that these transport phenomena are accessible to experiments.