This review article, authored by Peter Reimann, focuses on transport phenomena in spatially periodic systems far from thermal equilibrium, with a particular emphasis on directed transport in Brownian motors (ratchets). The main goal is to understand how directed transport can emerge in such systems, which are driven away from thermal equilibrium by non-equilibrium perturbations. The article covers a wide range of topics, including the basic concepts, historical landmarks, and various models of ratchets. Key points include: 1. **Basic Concepts and Phenomena**: The article introduces the Smoluchowski-Feynman ratchet, a thought experiment where a ratchet and pawl system is designed to convert unbiased Brownian motion into useful work. However, it is shown that no preferential direction of motion can be achieved due to the second law of thermodynamics. 2. **Historical Landmarks**: The history of the development of the ratchet effect is reviewed, highlighting key contributions from Smoluchowski, Feynman, Brillouin, and others. The article also discusses the experimental realization of ratchet effects in molecular motors and the Seebeck effect. 3. **Modeling and Analysis**: The article presents a simplified stochastic model of a Brownian particle in a periodic potential with broken spatial symmetry. It derives the Fokker-Planck equation and discusses the particle current, showing that even with broken symmetry, no systematic preferential motion occurs. 4. **General Framework**: The article outlines the general framework for ratchet models, including the role of symmetries, current inversions, and asymptotic regimes. 5. **Pulsating and Tilted Ratchets**: Detailed discussions on pulsating ratchets, where the potential shape varies over time, and tilted ratchets, where an additional static force is applied, are provided. 6. **Experimental Realizations and Biological Applications**: The article reviews experimental realizations of ratchets and their applications in biological systems, such as molecular motors and pumps. 7. **Extensions and Applications**: The review covers extensions to quantum mechanical ratchets, collective effects in interacting ratchets, and the influence of spatial disorder. 8. **Conclusion**: The article concludes with a summary of the main findings and future perspectives, emphasizing the importance of ratchet effects in both theoretical and practical contexts. Overall, the review provides a comprehensive overview of the field of Brownian motors, from fundamental concepts to advanced theoretical and experimental results.This review article, authored by Peter Reimann, focuses on transport phenomena in spatially periodic systems far from thermal equilibrium, with a particular emphasis on directed transport in Brownian motors (ratchets). The main goal is to understand how directed transport can emerge in such systems, which are driven away from thermal equilibrium by non-equilibrium perturbations. The article covers a wide range of topics, including the basic concepts, historical landmarks, and various models of ratchets. Key points include: 1. **Basic Concepts and Phenomena**: The article introduces the Smoluchowski-Feynman ratchet, a thought experiment where a ratchet and pawl system is designed to convert unbiased Brownian motion into useful work. However, it is shown that no preferential direction of motion can be achieved due to the second law of thermodynamics. 2. **Historical Landmarks**: The history of the development of the ratchet effect is reviewed, highlighting key contributions from Smoluchowski, Feynman, Brillouin, and others. The article also discusses the experimental realization of ratchet effects in molecular motors and the Seebeck effect. 3. **Modeling and Analysis**: The article presents a simplified stochastic model of a Brownian particle in a periodic potential with broken spatial symmetry. It derives the Fokker-Planck equation and discusses the particle current, showing that even with broken symmetry, no systematic preferential motion occurs. 4. **General Framework**: The article outlines the general framework for ratchet models, including the role of symmetries, current inversions, and asymptotic regimes. 5. **Pulsating and Tilted Ratchets**: Detailed discussions on pulsating ratchets, where the potential shape varies over time, and tilted ratchets, where an additional static force is applied, are provided. 6. **Experimental Realizations and Biological Applications**: The article reviews experimental realizations of ratchets and their applications in biological systems, such as molecular motors and pumps. 7. **Extensions and Applications**: The review covers extensions to quantum mechanical ratchets, collective effects in interacting ratchets, and the influence of spatial disorder. 8. **Conclusion**: The article concludes with a summary of the main findings and future perspectives, emphasizing the importance of ratchet effects in both theoretical and practical contexts. Overall, the review provides a comprehensive overview of the field of Brownian motors, from fundamental concepts to advanced theoretical and experimental results.