  The dynamics of quantum systems unfolds within a subspace of the state space
or operator space, known as the Krylov space. This review presents the use of
Krylov subspace methods to provide a compact and computationally efficient
description of quantum evolution, with emphasis on nonequilibrium phenomena of
many-body systems with a large Hilbert space. It provides a comprehensive
update of recent developments, focused on the quantum evolution of operators in
the Heisenberg picture as well as pure and mixed states. It further explores
the notion of Krylov complexity and associated metrics as tools for quantifying
operator growth, their bounds by generalized quantum speed limits, the
universal operator growth hypothesis, and its relation to quantum chaos,
scrambling, and generalized coherent states. A comparison of several
generalizations of the Krylov construction for open quantum systems is
presented. A closing discussion addresses the application of Krylov subspace
methods in quantum field theory, holography, integrability, quantum control,
and quantum computing, as well as current open problems.
