This review by Haye Hinrichsen focuses on nonequilibrium statistical physics, particularly phase transitions from fluctuating phases into absorbing states. The universality class of directed percolation is investigated in detail, covering various lattice models, scaling properties, field-theoretic aspects, numerical techniques, and experimental realizations. The review also discusses other classes of absorbing-state transitions not belonging to the directed percolation universality class and the concept of damage spreading, which is shown to be ambiguous and unable to define chaotic and regular phases in stochastic nonequilibrium systems. Additionally, the review explores depinning transitions in models of interface growth related to phase transitions into absorbing states. The content is structured into several sections, including an introduction to stochastic many-particle systems, directed percolation, other classes of spreading transitions, damage spreading, and interface growth. The review aims to provide a comprehensive overview of recent developments in the field, highlighting the rich and complex behavior of nonequilibrium systems.This review by Haye Hinrichsen focuses on nonequilibrium statistical physics, particularly phase transitions from fluctuating phases into absorbing states. The universality class of directed percolation is investigated in detail, covering various lattice models, scaling properties, field-theoretic aspects, numerical techniques, and experimental realizations. The review also discusses other classes of absorbing-state transitions not belonging to the directed percolation universality class and the concept of damage spreading, which is shown to be ambiguous and unable to define chaotic and regular phases in stochastic nonequilibrium systems. Additionally, the review explores depinning transitions in models of interface growth related to phase transitions into absorbing states. The content is structured into several sections, including an introduction to stochastic many-particle systems, directed percolation, other classes of spreading transitions, damage spreading, and interface growth. The review aims to provide a comprehensive overview of recent developments in the field, highlighting the rich and complex behavior of nonequilibrium systems.