This book introduces integrable peaked soliton (peakon) systems, which has been used by mathematical analysists and theoretical physicists to study the Camassa-Holm (CH) equation from various points and extended to create peakon models in nonlinear science. Recently integrable peakon models have been designed that are scalar and vector forms of the peakon systems. Quadratic and cubic peakon models as well as multi-component peakon dynamical systems, new multi-peakon/kink interactions, Hamiltonian systems, Lax pairs, and applications in nonlinear sciences are detailed and explained with examples to aid scientific scholars to conduct theoretical and applied research projects. Advanced undergraduate and graduate students in applied mathematics, physics and engineering can use this book for a course in integrable peakon systems. The technical aspect of peakon solution will be useful for graduate students and researchers in mathematics, theoretical physics, engineering, nonlinear science and other related fields. Researchers will find the techniques and complicated simulation procedure explored in this book vital for advancing their individual research.