  Matrix functions are a central topic of linear algebra, and problems
requiring their numerical approximation appear increasingly often in scientific
computing. We review various limited-memory methods for the approximation of
the action of a large-scale matrix function on a vector. Emphasis is put on
polynomial methods, whose memory requirements are known or prescribed a priori.
Methods based on explicit polynomial approximation or interpolation, as well as
restarted Arnoldi methods, are treated in detail. An overview of existing
software is also given, as well as a discussion of challenging open problems.
