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Emergent many-body phenomena are at the core of the exciting properties of strongly correlated materials. Understanding them requires confronting the many-body problem. While, at first, this appears to be an impossible task, substantial progress has been made by combining physical insights with modern numerical approaches. A successful strategy is to devise methods that use the understanding gained from simple models for the construction of physically motivated wave-functions. Results for the ground state of real materials can then be obtained by optimizing them via deterministic or stochastic algorithms. The methods of choice for determining spectra are instead based on Green functions. The key idea is to map the complex realistic many-body Hamiltonian to a simpler auxiliary model that can be solved numerically.