Fundamental concepts pertaining to the solution of heat conduction problems by probability methods (Monte Carlo methods) are devised and techniques of application are developed. It is demonstrated that probability methods can be applied over the entire range of heat conduction problems. These include both steady state and transient situations in bodies of arbitrary shape with arbitrary boundary conditions (including derivative conditions) and with volume heat sources. Problems involving nonlinear boundary conditions (e.g., combined convection and radiation) and moving boundaries (change of phase) are also accommodated by probability methods. All of these various situations are treated in the paper. Numerous computational experiments are carried out, many of which provide results for physical problems not heretofore solved in the literature. A new concept, the floating random walk, is introduced and this provides the flexibility needed to accommodate such a wide range of problems.