This book aims to provide an overview of the special functions of fractional calculus and their applications in diffusion and random search processes. The book contains detailed calculations for various examples of anomalous diffusion, random search and stochastic resetting processes, which can be easily followed by the reader, who will be able to reproduce the obtained results. The book will be intended for advanced undergraduate and graduate students and researchers in physics, mathematics and other natural sciences due to the various examples which will be provided in the book. Sample Chapter(s) Preface Chapter 1: Mathematical background Contents: Mathematical Background Fox H-Function and Related Functions Elements of Random Walk Theory CTRW on Combs Heterogeneous Diffusion Processes Diffusion Processes with Stochastic Resetting Random Search Diffusion on Fractal Tartan Finite-Velocity Diffusion Appendices: Functional Calculus Stochastic Differential Equations Large Deviation Principle Fractals and Fractal Dimension Implementation of Wolfram Mathematica Readership: Advanced undergraduate and graduate students, researchers in the fields of stochastic processes, fractional calculus, anomalous diffusion, random search, stochastic resetting, econophysics.