The intricate relationship between the structure and function of physical transport networks remains to large degree a mystery, despite extensive statistical classification of the various graph phenotypes. Unlike mathematical constructs on paper, physical networks live in real space, adapt to noisy environments, and provide efficient transport pathways. The evidence for this is overwhelming throughout the evolutionary arena of life: transport networks are present from intracellular level to the complex vasculatures of mammals. This thesis posits that the transport processes occurring within networks fundamentally influence the network architecture. In this thesis, we adopt a physics approach to explore the interplay between spatial embedding, network topology, and ubiquitous transport mechanisms like diffusion. Our primary aim is to decipher how spatial constraints affect the optimal transport architectures. Spatial embedding can introduce time delays in the case of diffusive transport within links where the time to traverse an edge scales non-linearly with time. Contrary to intuition, adding topological shortcuts between distant nodes can worsen the overall search efficiency of the network and conversely, edge removal can improve transport. Using optimization we reveal a crossover in the optimal search architecture ranging from dense graphs for super-diffusion to sparse for sub-diffusion. Further, we extend our study beyond Euclidean space to describestochastic transitions between the metastable states in an energy landscape where nodes represent energy minima and edges act as transition pathways with associated energy barriers that introduce exponential waiting times to traverse them. The goal here is to analyze how the landscape's topology affects the search efficiency for the global minimum. This is critical for processes like protein folding. Finally, we explore how spatial constraints introduce heterogeneities in supply networks: as nutrients travel through the microcirculation and are absorbed, their availability continuously decreases. We propose a theory for vessel adaptation, inspired by the critical role of oxygen delivery, that balances solute distribution, material cost, and hydrodynamic resistance leading to networks that combine features of both hierarchical trees and uniform meshes. This trade-off is mainly controlled by the vessels' metabolism. Surprisingly, by comparing the optimal architectures with experimentally measured networks in the rat mesentery we can extrapolate the oxygen absorption rate that agrees with independent in vivo measurements.