  This review is devoted to the detailed consideration of the universal
statistical properties of one-dimensional directed polymers in a random
potential. In terms of the replica Bethe ansatz technique we derive several
exact results for different types of the free energy probability distribution
functions. In the second part of the review we discuss the problems which are
still waiting for their solutions. Several mathematical appendices in the
ending part of the review contain various technical details of the performed
calculations.
