This book aims to provide the basic theory of fractional calculus and its applications based on practical schemes and approaches, illustrated with applicable engineering and technical examples, especially focusing on the fractional-order controller design. This expanded second edition bridges rigorous fractional calculus theory with cutting-edge engineering applications. It offers new case studies (e.g., stabilization control of a general class of fractional-order systems, fractional-order recurrent neural dynamical systems, etc.), enhanced MATLAB® implementations, and updated numerical methods for fractional-order controller design. Ideal for researchers and engineers, the book demystifies complex concepts like Grünwald-Letnikov and Caputo derivatives through real-world examples in robotics, chatter suppression, and engine control. In addition, this book: Theory-to-practice focus: Unifies mathematical foundations (proofs, properties) with implementable controller designs MATLAB® integration: Provides code for fractional-order PID tuning and numerical simulations Unique case studies: Including multilink robots, rotary inverted pendulums, and ICE idle speed control.