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Eukaryotic cells contain organelle systems that organize and distribute molecular components. In this talk we will explore two such systems: mitochondria and the endoplasmic reticulum (ER), both of which form extensively networked architectures that span across the cell. While functionally distinct, both of these organelles give rise to fundamental physical questions: How are network structures formed and maintained? How does the architecture of the network modulate its ability to transport proteins and ions through the cell interior? In the case of mitochondria, morphologies ranging from highly fragmented `social' networks to highly branched tubular networks are achieved through fusion and fission. By contrast, we show the structure and dynamics of the peripheral ER is well-approximated by a liquid network model that incorporates tension and tubular outgrowth. Using these dynamic network models, we explore the rate of material transport in different mitochondrial network structures and in the interconnected tubular mesh of the ER, highlighting the key physical features that govern the rate of dispersion. Theoretical results are compared against experimental data on the spreading of photoconverted proteins within the organelles. Finally, we demonstrate a key functional role for highly connected reticulated architectures in facilitating calcium transfer between the ER and mitochondria.