This book introduces in a constructive manner a general framework for regression and fitting methods for many applications and tasks involving data on manifolds. The methodology has important and varied applications in machine learning, medicine, robotics, biology, computer vision, human biometrics, nanomanufacturing, signal processing, and image analysis, etc. The first chapter gives  motivation examples, a wide range of applications, raised challenges,  raised challenges, and some concerns.  The second chapter gives a comprehensive exploration and step-by-step illustrations for Euclidean cases. Another dedicated chapter covers  the geometric tools needed for each manifold and provides expressions and key notions for any application for manifold-valued data.  All loss functions and optimization methods are given as algorithms and can be easily implemented. In particular, many popular manifolds are considered with  derived and specific formulations. The same philosophy is used in all chapters and all novelties are illustrated with intuitive examples. Additionally, each chapter includes simulations and experiments  on real-world problems for understanding and potential extensions for a wide range of applications.