Methods for solving one-dimensional direct and inverse heat conduction problems based on the Duhamel integral are the subject of in this chapter. The superposition method for time-dependent thermal boundary conditions is presented in Sect. 4.1. The Duhamel integral and its application to solving linear transient heat conduction problems for time-dependent boundary conditions are addressed in Sect. 4.2. The superposition method and Duhamel integral are often used to generate “accurate measurement data” and verify solutions to inverse problems of transient heat conduction. It should be emphasised that both methods mentioned can be applied to linear problems when the physical properties of the material and the boundary conditions are independent of temperature. Applying the superposition method, it is possible, for example, to find a close exact solution for a plate with constant thermophysical properties heated by a heat flux varying in time in the form of a triangular or rectangular pulse. The Stolz and Beck methods for solving IHCPs, described in Sect. 4.3, are based on the Duhamel integral. In this chapter contains many examples illustrating the use of the superposition method and the Duhamel integral to solve direct problems and the Stolz and Beck method to solve inverse heat conduction problems.