The term diffusion refers to the mixing of different substances like gases or liquids, based on the thermal motion of its individual particles, such as the process by which a drop of ink spreads in a glass of water over time. This process of mixing (the Latin word diffundere means to scatter or to spread out) can be described on a macroscopic level by Fick’s equations. The physical description of diffusion connects macroscopically visible phenomena with the molecular kinetic theory of heat. This connection, placed on a solid theoretical foundation by Einstein (Annalen der Physik, 322(8):549–560, 1905), was an important argument for the correctness of the atomic theory. This view explained that the visible movement of individual pollen particles in a water bath, as described before by Brown and others (Philosophical Magazine Series 2, 4(21):161–173, 1828); Gouy (J Phys Theor Appl, 7(1):561–564, 1888); Ingenhousz and Molitor (Vermischte Schriften physisch-medicinischen Inhalts. Gedruckt und verlegt bey Christian Friedrich Wappler, Wien, 1784), is due to the impact of colliding water molecules. In the following sections, we will first discuss some biological examples where diffusion plays an important role, and then, based on considerations about the situation of a single molecule in a discrete lattice model, illustrate the fundamental diffusion equations and thus bridge to the previously discovered macroscopic diffusion equations. Thermal diffusion sets an upper limit for the passive transport speed of molecules, and we will briefly discuss its implications in biological systems and how nature has found other solutions for faster transport.