Gravitational waves offer a unique way to probe the early Universe, as they can travel through the cosmos without being absorbed. This article summarizes the prospects for detecting gravitational wave backgrounds and the various primordial events that could generate such signals. It discusses the detection methods, including pulsar timing arrays, astrometry, and interferometers, and explores the sources of gravitational waves from cosmic phase transitions, topological defects, and scalar-induced gravitational waves. The article also covers other significant sources such as inflation and reheating, and concludes with the potential of gravitational wave cosmology to address fundamental questions about the early Universe. The review highlights the importance of future detectors in uncovering the first second of the Universe's history and the challenges in modeling primordial gravitational wave sources.Gravitational waves offer a unique way to probe the early Universe, as they can travel through the cosmos without being absorbed. This article summarizes the prospects for detecting gravitational wave backgrounds and the various primordial events that could generate such signals. It discusses the detection methods, including pulsar timing arrays, astrometry, and interferometers, and explores the sources of gravitational waves from cosmic phase transitions, topological defects, and scalar-induced gravitational waves. The article also covers other significant sources such as inflation and reheating, and concludes with the potential of gravitational wave cosmology to address fundamental questions about the early Universe. The review highlights the importance of future detectors in uncovering the first second of the Universe's history and the challenges in modeling primordial gravitational wave sources.