Markov process theory is basically an extension of ordinary calculus to accommodate functions whose time evolutions are not entirely deterministic. It is a subject that is becoming increasingly important for many fields of science. This book develops the single-variable theory of both continuous and jump Markov processes in a way that should appeal especially to physicists and chemists at the senior and graduate level. It offers a self-contained, pragmatic exposition of the needed elements of random variable theory. It includes logically integrated derviations of the Chapman-Kolmogorov equation, the Kramers-Moyal equations, the Fokker-Planck equations, the Langevin equation, the master equations, and the moment equations. It contains a detailed exposition of Monte Carlo simulation methods, with plots of many numerical examples. It presents clear treatments of first passages, first exits, and stable state fluctuations and transitions. It features carefully drawn applications to Brownian motion, molecular diffusion, and chemical kinetics.