This textbook is an introduction to the Brownian motion of colloids and nano-particles, and the diffusion of molecules. One very appealing aspect of Brownian motion, as this book illustrates, is that the subject connects a broad variety of topics, including thermal physics, hydrodynamics, reaction kinetics, fluctuation phenomena, statistical thermodynamics, osmosis and colloid science. The book is based on a set of lecture notes that the authors used for an undergraduate course at the University of Utrecht, Netherland. It aims to provide more than a simplified qualitative description of the subject, without getting bogged down in difficult mathematics. Each chapter contains exercises, ranging from straightforward ones to more involved problems, addressing instances from (thermal motion in) chemistry, physics and life sciences. Exercises also deal with derivations or calculations that are skipped in the main text. The book offers a treatment of Brownian motion on a level appropriate for bachelor/undergraduate students of physics, chemistry, soft matter and the life sciences. PhD students attending courses and doing research in colloid science or soft matter will also benefit from this book.