Brownian diffusion, the motion of large molecules in a sea of very many much smaller molecules, is topical because it is one of the ways in which biologically important molecules move about inside living cells. This book presents the mathematical physics that underlies the four simplest models of Brownian diffusion. In a lively tutorial style, the authors discuss some of the most widely used mathematical formulations of diffusion. They have endeavored to organize and present the subject matter "from a purely logical perspective". They emphasize the basic physical assumptions and the conditions for the validity of each of the mathematical formalisms. No subtlety is bypassed, and no limitation of the theory is swept under the carpet.