Materials can harbour quantum many-body systems, most typically in the form of strongly correlated electrons in solids, that lead to novel and remarkable functions thanks to emergence-collective behaviours that arise from strong interactions among the elements. These include the Mott transition, high-temperature superconductivity, topological superconductivity, colossal magnetoresistance, giant magnetoelectric e ect, and topological insulators. These phenomena will probably be crucial for developing the next-generation quantum technologies that will meet the urgent technological demands for achieving a sustainable and safe society. Dissipationless electronics using topological currents and quantum spins, energy harvesting such as photovoltaics and thermoelectrics, and secure quantum computing and communication are the three major fields of applications working towards this goal. Here, we review the basic principles and the current status of the emergent phenomena and functions in materials from the viewpoint of strong correlation and topology.E mergence is a concept developed for many-body systems, indicating the properties, phenomena, and functions that never appear in the individual elements but are realized only when a huge number of elements get together 1 . Inside materials, emergent phenomena are often seen due to the interaction of electronic states; with recent developments on electronic states in solids schematically summarized in Fig. 1a. Electrons in solids have several degrees of freedom: charge, spin and orbital, and are characterized by the topological nature determined by the atomic potential on the crystal lattice structure, as schematically shown in Fig. 1b. These five attributes are coupled together and determine the overall responses to stimuli, which appear as materials' electrical, magnetic, optical, thermal, and mechanical properties.These collective electrons demonstrate various macroscopic quantum phenomena, the representative example of which is superconductivity. One of the big challenges in condensed matter physics is to increase the temperature (T c ) at which superconductivity can be observed to above room temperature. However, there are many other macroscopic quantum phenomena that are already observable above room temperature. One is magnetism (by the Bohr-van Leeuwen theorem), and the other is ferroelectricity 2,3 . Remarkably, there are common features of these three macroscopic quantum phenomena: the order parameter, which is of quantum origin, behaves as a classical quantity, and the quantum topology plays a crucial role.The topological nature of the electronic states is the key concept in the most recent developments in understanding quantum materials. The quantum Hall effect, topological insulators, and topological superconductors are all characterized by nontrivial topologies in Hilbert space. The Berry phase, which describes the connection and curvature of the subspace of Hilbert space, plays the central role in the unified principle to describe this topological nature 4 . ...