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, focusing on the quantum evolution of operators in the Heisenberg picture as well as pure and mixed states. The paper 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. It also compares several generalizations of the Krylov construction for open quantum systems. The paper discusses the application of Krylov subspace methods in quantum field theory, holography, integrability, quantum control, and quantum computing, as well as current open problems. Krylov subspace methods are essential in scientific computing, projecting high-dimensional problems onto lower-dimensional Krylov subspaces to make them more tractable. These methods are particularly useful for solving large-scale linear algebra problems, which are common in science and engineering. They are especially efficient for sparse or structured matrices where direct methods are computationally impractical. Krylov subspace methods have become increasingly relevant in the study of classical and quantum many-body systems, where they are also known as the recursion method. For quantum systems, the time evolution is described by a trajectory of the quantum state in Hilbert space. Krylov subspace methods offer a powerful approach by identifying the minimal subspace in which the dynamics unfolds, without the need to fully diagonalize the Hamiltonian and explicitly store the quantum state. Their formulation in the Heisenberg evolution is also frequent in this context and has long proved useful in the study of correlation functions, linear response theory, spectral functions, and other equilibrium properties. The study of quantum dynamics in Krylov space is largely motivated by progress in understanding quantum chaos. Much of the background has been reviewed in excellent references. We provide only a succinct account with emphasis on recent developments, discussing a selection of measures to diagnose quantum chaos that are relevant to the study of quantum dynamics in Krylov space. Random matrix theory (RMT) finds broad applications in science and engineering. A series of landmark works by Wigner and Dyson introduced RMT in physics. This provided a way to describe the spectra of heavy nuclei and complex quantum systems with minimal information about the underlying Hamiltonian. In doing so, Dyson identified the role played by the symmetries of the system, introducing a classification known as the three-fold way. The use of RMT in quantum physics was further spurred by the study of chaos across that quantum-to-classical transition. The Bohigas-Giannoni-Schmit (BGS) conjecture posits that the spectral statistics of quantum chaotic systems are described by RMT, while generic integrable systems show no correlations in the spectrumThis 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, focusing on the quantum evolution of operators in the Heisenberg picture as well as pure and mixed states. The paper 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. It also compares several generalizations of the Krylov construction for open quantum systems. The paper discusses the application of Krylov subspace methods in quantum field theory, holography, integrability, quantum control, and quantum computing, as well as current open problems. Krylov subspace methods are essential in scientific computing, projecting high-dimensional problems onto lower-dimensional Krylov subspaces to make them more tractable. These methods are particularly useful for solving large-scale linear algebra problems, which are common in science and engineering. They are especially efficient for sparse or structured matrices where direct methods are computationally impractical. Krylov subspace methods have become increasingly relevant in the study of classical and quantum many-body systems, where they are also known as the recursion method. For quantum systems, the time evolution is described by a trajectory of the quantum state in Hilbert space. Krylov subspace methods offer a powerful approach by identifying the minimal subspace in which the dynamics unfolds, without the need to fully diagonalize the Hamiltonian and explicitly store the quantum state. Their formulation in the Heisenberg evolution is also frequent in this context and has long proved useful in the study of correlation functions, linear response theory, spectral functions, and other equilibrium properties. The study of quantum dynamics in Krylov space is largely motivated by progress in understanding quantum chaos. Much of the background has been reviewed in excellent references. We provide only a succinct account with emphasis on recent developments, discussing a selection of measures to diagnose quantum chaos that are relevant to the study of quantum dynamics in Krylov space. Random matrix theory (RMT) finds broad applications in science and engineering. A series of landmark works by Wigner and Dyson introduced RMT in physics. This provided a way to describe the spectra of heavy nuclei and complex quantum systems with minimal information about the underlying Hamiltonian. In doing so, Dyson identified the role played by the symmetries of the system, introducing a classification known as the three-fold way. The use of RMT in quantum physics was further spurred by the study of chaos across that quantum-to-classical transition. The Bohigas-Giannoni-Schmit (BGS) conjecture posits that the spectral statistics of quantum chaotic systems are described by RMT, while generic integrable systems show no correlations in the spectrum