Explore emergent network geometry in this 43-minute lecture by Ginestra Bianconi, presented as part of the Hausdorff Trimester Program on Applied and Computational Algebraic Topology. Delve into topics such as network topology, clique percolation, world association networks, and brain functional topology. Examine collaboration networks, hidden metrics of complex networks, and quantum spacetime. Investigate quantum gravity approaches to networks and simplicial complexes, scale-free networks, and the unitary cell of growing networks. Learn about fitness, energy distribution in Cayley tree networks, attachment probability, and generalized degree distributions. Conclude with insights on discrete manifolds and related processes, gaining a comprehensive understanding of the intersection between network science and geometry.