This book offers a concise presentation of theoretical concepts characterizing and quantifying the slowing down of swift heavy ions in matter. Although the penetration of charged particles through matter has been studied for almost a hundred years, the quantitative theory for swift penetrating ions heavier than helium has been developed mainly during the past decade and is still progressing rapidly. The book addresses scientists and engineers working at accelerators with an interest in materials analysis and modification, medical diagnostics and therapy, mass spectrometry and radiation damage, as well as atomic and nuclear physicists. Although not a textbook, this monograph represents a unique source of state-of-the-art information that is useful to a university teacher in any course involving the interaction of charged particles with matter.