This master thesis serves as a support for the Computational Continuum Mechanics course for advanced undergraduates and first-year graduate students at Montanuniversitaet Leoben. It discusses the fundamentals of heat transfer phenomenon, and the analytical and numerical approaches to heat conduction problems are reviewed. The first problem introduces steady state heat conduction in two-dimensional (2D) slab. In order to obtain a numerical solution, the step-by-step description of this case in OpenFOAM is given. Then the results of the OpenFOAM case are compared to theoretical calculations, and possible reasons for the existing errors are noticed. The second example examines the one dimensional (1D) steady state heat conduction problem in terms of cylinder insulation. This part describes the effect of critical insulation radius in case of non-planar geometries when a hot thin pipe is cooled down to an ambient temperature. The detailed set up with all codes to run a successful case are presented. At the end of this chapter the comparison between the analytical and numerical solution is performed. Dealing with 1D transient heat conduction problem, quenching a steel plate process is taken as the third case. As with the previous cases, the theoretical and computational solutions are shown. Further investigation of the problem offers the explanation of certain errors in the numerical solution. The last part, heat transfer in a complex three-dimensional (3D) figure, is simulated in order to show that in spite of the complicated geometry a problem shall be approached in the same manner as a problem in simpler forms.