Data assimilation is an approach that combines observations and model output, with the objective of improving the latter. This book places data assimilation into the broader context of inverse problems and the theory, methods, and algorithms that are used for their solution. It provides a framework for, and insight into, the inverse problem nature of data assimilation, emphasising 'why' and not just 'how'. Methods and diagnostics are emphasised, enabling readers to readily apply them to their own field of study. This comprehensive guide is accessible to non-experts and contains numerous examples and diverse applications from a broad range of domains, including geophysics, environmental acoustics, medical imaging, mechanical and biomedical engineering, economics and finance, and traffic control and urban planning. Readers will also find included the latest methods for advanced data assimilation, combining variational and statistical approaches.