  This preprint is of a chapter to appear in {\it Combinatorics and finite
fields: Difference sets, polynomials, pseudorandomness and applications. Radon
Series on Computational and Applied Mathematics}, K.-U. Schmidt and A.
Winterhof (eds.).
  Rank-metric codes are codes consisting of matrices with entries in a finite
field, with the distance between two matrices being the rank of their
difference. Codes with maximum size for a fixed minimum distance are called
Maximum Rank Distance (MRD) codes. Such codes were constructed and studied
independently by Delsarte (1978), Gabidulin (1985), Roth (1991), and
Cooperstein (1998). Rank-metric codes have seen renewed interest in recent
years due to their applications in random linear network coding.
  MRD codes also have interesting connections to other topics such as
semifields (finite nonassociative division algebras), finite geometry,
linearized polynomials, and cryptography. In this chapter we will survey the
known constructions and applications of MRD codes, and present some open
problems.
