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Mathematical and Quantum-Mechanical Background.- General Concepts of the Theory of Resonance States and Processes.- Theory of Resonance States Based on the Hilbert-Schmidt Expansion.- Projection Methods.- Theory of Resonance States and Processes Based on Analytical Continuation in the Coupling Constant.- S-matrix Parametrization of Scattering Data. Extraction of Resonance Parameters from Experimental Data.- Resonances in Atomic Physics.- Conclusion, Open Problems.