Random walks of molecular motors, which bind to and unbind from cytoskeleta
l filaments, are studied theoretically. The bound and unbound motors underg
o directed and nondirected motion, respectively. Motors in open compartment
s exhibit anomalous drift velocities. Motors in closed compartments generat
e 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. Quantitat
ive estimates based on experimental data for bound motors indicate that the
se transport phenomena are accessible to experiments.