A new modification of the McEliece public-key cryptosystem is proposed that employs the so-called maximum-rank-distance (MRD) codes in place of Goppa codes and that hides the generator matrix of the MRD code by addition of a randomly-chosen matrix. A short review of the mathematical background required for the construction of MRD codes is given. The cryptanalytic work function for the modified McEliece system is shown to be much greater than that of the original system. Extensions of the rank metric are also considered.