This book is concerned with the fault estimation problem for network systems. Firstly, to improve the existing adaptive fault estimation observer, a novel so-called intermediate estimator is proposed to identify the actuator or sensor faults in dynamic control systems with high accuracy and convergence speed. On this basis, by exploiting the properties of network systems such as multi-agent systems and large-scale interconnected systems, this book introduces the concept of distributed intermediate estimator; faults in different nodes can be estimated simultaneously; meanwhile, satisfactory consensus performances can be obtained via compensation based protocols. Finally, the characteristics of the new fault estimation methodology are verified and discussed by a series of experimental results on networked multi-axis motion control systems. This book can be used as a reference book for researcher and designer in the field of fault diagnosis and fault-tolerant control and can also be used as a reference book for senior undergraduate and graduate students in colleges and universities.