The dynamics of quantum systems typically unfolds within a subspace of the state or operator space, known as the Krylov space. Krylov subspace methods provide a compact, computationally efficient description of quantum evolution and quantum chaos, particularly for nonequilibrium phenomena in many-body systems with large Hilbert spaces. In this talk, I will discuss a notion of complexity defined in Krylov space as a probe of operator growth, quantum chaos, information scrambling, and quantum state evolution in both closed and open quantum systems. Illustrative examples include quantum spin chains, Sachdev-Ye-Kitaev models, and random matrix theory. The talk is based on selected topics from the review article arXiv:2405.09628 (Physics Reports).