This chapter reviews the fundamental physics associated with the penetration of energetic ions into solids. It describes the basic physics of how the ions lose energy to the solid, creating damage, and the final distribution of these ions after they stop within the solid. The phenomenology of ion implantation damage and annealing is covered in the chapters on Radiation Damage and Annealing. For extended reading in this subject, see (Ziegler and Biersack, “SRIM—the stopping and range of lons in matter”, LuLu Press www.SRIM.org ). In this chapter, we first present the concepts of finding the distributions of ion implantation as calculated by the computer program SRIM (The Stopping and Range of lons in Matter) which is freeware ( www.SRIM.org ). The physics of ion penetration of matter is then presented in an historical context with a review of the 90 years of stopping and range theory so it can be seen how various aspects became important because of specific scientific interests in atomic theory, quantum mechanics, radioactive atoms, nuclear physics and ion implantation. Then a detailed presentation is made of the physics of ion interactions with matter. The details are presented of the electronic excitation of the target atoms, the displacement of lattice atoms by energetic collisions (lattice damage) and the production of plasmons and phonons in the solid by the passing ions. All of the above are discussed in the SRIM textbook (Ziegler and Biersack, “SRIM—the stopping and range of lons in matter”, LuLu Press www.SRIM.org ).