This book provides a comprehensive review of the present status of achievements in the area of soft X-ray laser sources, supplemented by information about sources based on relativistic laser˗matter interaction and their future, and incoherent sources within a very broad spectral range. The diversity of demonstrated or planned applications presented in the book supports the thesis that such sources have now reached a mature stage of development. There is a significant effort worldwide to develop very bright, ultra-short duration, radiation sources in the extreme ultraviolet and X-ray spectral regions, driven by a diversity of potential applications in nearly all branches of science. This book updates the status in this field and focuses on developments in laser plasma-based methods. The scheme of transient inversion proves its robustness by being dominant in the area of repetitive X-ray lasers pumped at grazing-incidence-geometry by optical lasers of moderate energy at increasing repetition rates – these characteristics enable them to be used in university-class laboratories. Experimental and theoretical progress associated with the seeding technique by high harmonics is well covered. This technique is important as it is also considered from the perspective of seeding X-ray free electron lasers. Dramatic progress in laser development and extending the available power range to the petawatt level is stimulating a new generation of methods of ultra-bright and ultra-short pulses of energetic X-rays. These methods, as well as some elements of attoscience based on high harmonics generation, are well represented. Finally, a comprehensive survey of possible applications of the described sources in different areas of science constitutes, by defining the parameter set of interest, a framework for considering the direction further developments will take.