This thesis is a report of the work I have done over the last five years to study thermodynamic and kinetic aspects of single-molecule behavior in the condensed phase. It is concerned specifically with the development of analytically tractable models of various phenomena that have been observed in experiments on such single-molecule systems as colloids, double-stranded DNA, multi-unit proteins, and enzymes. In fluid environments, the energetics, spatial conformations, and chemical reactivity of these systems undergo fluctuations that can be characterized experimentally in terms of time correlation functions, survival probabilities, mean first passage times, and related statistical parameters. The thesis shows how many of these quantities can be calculated in closed form from a model based on simple Brownian motion, or generalizations of it involving fractional calculus. The theoretical results obtained here have been shown to agree qualitatively or quantitatively with a range of experimental data. The thesis therefore demonstrates the effectiveness of Brownian motion concepts as a paradigm of stochasticity in biological processes.