Lyapunov analysis deals with the behavior of small perturbations of trajectories of general dynamical systems. Although the growth or decay strength of such perturbations, as described by the spectrum of Lyapunov exponents, has been investigated in detail, it was only recently that the associated directions in state space, the Lyapunov vectors, came into the focus of the scientific community. They are the generalization of normal modes of harmonic systems to nonlinear, chaotic systems. The interest was triggered by the observation that some of these vectors may exhibit long wavelength and low-frequency behavior similar to that of classical hydrodynamic modes. In this chapter, we discuss major aspects of these hydrodynamic Lyapunov modes (HLM) for Hamiltonian and dissipative systems, and also more recent findings about the role of so-called covariant Lyapunov vectors (CLV) in determining the effective number of degrees of freedom of extended systems.