In this paper, we review the main advances made by our research group on the heat transfer performance of complex flow architectures embedded in a conducting solid. The immediate applications of this work include the design of ground-coupled heat pumps, seasonal thermal energy storage systems, and district heating and cooling systems. Various configurations are considered: U-shaped ducts with varying spacing between the parallel portions of the U, serpentines with three elbows, and trees with T-shaped and Y-shaped bifurcations. In each case, the volume ratio of fluid to soil is fixed. We found the critical geometric features that allow the heat transfer density of the stream-solid configuration to be the highest. In the case of U-tubes and serpentines, the best spacing between parallel portions is discovered, whereas the vascular designs morph into bifurcations and angles of connection that provide progressively greater heat transfer rate per unit volume. We show that the flow of heat into or out of a solid volume must have an S-shaped history curve that is entirely deterministic. This constructal-design principle unites a wide variety of previously disconnected S-curve phenomena (ground heat storage and retrieval, population growth, cancer, chemical reactions, contaminants, languages, news, information, innovations, technologies, economic activity).