This review addresses recent developments in nonequilibrium statistical physics, focusing on phase transitions into absorbing states. It investigates the directed percolation (DP) universality class in detail, discussing various lattice models, their scaling properties, field-theoretic aspects, numerical techniques, and experimental realizations. It also explores other absorbing-state transitions not belonging to the DP class and discusses damage spreading, a technique that is ambiguous in stochastic nonequilibrium systems. The review also covers depinning transitions in interface growth models related to phase transitions into absorbing states. The paper begins by introducing stochastic many-particle systems, including the one-dimensional random walk, master equation, asymmetric exclusion process, and reaction-diffusion processes. It discusses the mean field approximation, the influence of fluctuations, and numerical simulations. It then delves into directed percolation, explaining its role as a spreading process, lattice models, scaling theory, and field-theoretic formulation. It also covers surface critical behavior, quenched disorder, related models, and experimental realizations. The review then discusses other classes of spreading transitions, including long-range processes, absorbing-state transitions with additional symmetries, activated random walks, and self-organized criticality. It examines damage spreading, showing its ambiguity in defining chaotic and regular phases. The paper concludes with a discussion of interface growth, including roughening transitions, depinning transitions, and nonequilibrium wetting. The review highlights the importance of universality classes in nonequilibrium phase transitions, noting that they are governed by various symmetry properties. It emphasizes the robustness of the DP class and its significance in nonequilibrium statistical mechanics. The paper also discusses the challenges in solving nonequilibrium processes exactly and the need for approximation techniques. It concludes by emphasizing the importance of experimental studies in understanding nonequilibrium phase transitions.This review addresses recent developments in nonequilibrium statistical physics, focusing on phase transitions into absorbing states. It investigates the directed percolation (DP) universality class in detail, discussing various lattice models, their scaling properties, field-theoretic aspects, numerical techniques, and experimental realizations. It also explores other absorbing-state transitions not belonging to the DP class and discusses damage spreading, a technique that is ambiguous in stochastic nonequilibrium systems. The review also covers depinning transitions in interface growth models related to phase transitions into absorbing states. The paper begins by introducing stochastic many-particle systems, including the one-dimensional random walk, master equation, asymmetric exclusion process, and reaction-diffusion processes. It discusses the mean field approximation, the influence of fluctuations, and numerical simulations. It then delves into directed percolation, explaining its role as a spreading process, lattice models, scaling theory, and field-theoretic formulation. It also covers surface critical behavior, quenched disorder, related models, and experimental realizations. The review then discusses other classes of spreading transitions, including long-range processes, absorbing-state transitions with additional symmetries, activated random walks, and self-organized criticality. It examines damage spreading, showing its ambiguity in defining chaotic and regular phases. The paper concludes with a discussion of interface growth, including roughening transitions, depinning transitions, and nonequilibrium wetting. The review highlights the importance of universality classes in nonequilibrium phase transitions, noting that they are governed by various symmetry properties. It emphasizes the robustness of the DP class and its significance in nonequilibrium statistical mechanics. The paper also discusses the challenges in solving nonequilibrium processes exactly and the need for approximation techniques. It concludes by emphasizing the importance of experimental studies in understanding nonequilibrium phase transitions.