Fractional calculus is undergoing rapid and ongoing development. We can already recognize, that within its framework new concepts and strategies emerge, which lead to new challenging insights and surprising correlations between different branches of physics. This book is an invitation both to the interested student and the professional researcher. It presents a thorough introduction to the basics of fractional calculus and guides the reader directly to the current state-of-the-art physical interpretation. It is also devoted to the application of fractional calculus on physical problems, in the subjects of classical mechanics, friction, damping, oscillations, group theory, quantum mechanics, nuclear physics, and hadron spectroscopy up to quantum field theory. Sample Chapter(s) Foreword (106 KB) Chapter 1: Introduction (247 KB) Contents: Functions The Fractional Derivative Friction Forces Fractional Calculus The Fractional Harmonic Oscillator Wave Equations and Parity Nonlocality and Memory Effects Quantum Mechanics Fractional Spin: A Property of Particles Described with the Fractional Schrödinger Equation Factorization Symmetries The Fractional Symmetric Rigid Rotor q-Deformed Lie Algebras and Fractional Calculus Fractional Spectroscopy of Hadrons Higher Dimensional Fractional Rotation Groups Fractors: Fractional Tensor Calculus Fractional Fields Gauge Invariance in Fractional Field Theories Outlook Readership: Students and researchers in physics.