This course provides a thorough introduction into basic and advanced concepts in applied mathematics, mechanics and engineering (civil, mechanical and electrical) on the analysis and design of time-periodic systems. A balance is provided between classic procedures, recent trends in the analysis and experimental observations of resonant and non-resonant behavior. A wide range of problems and their dynamic aspects are covered starting from simple linear differential equations, over small perturbative) nonlinearities to strongly nonlinear systems including friction and contact, up to partial differential equations, delayed differential equations and large systems of differential equations. Special topics like high-frequency parametric excitation, quasi-periodic excitation, parametric excitation in gyroscopic systems and efficient analysis of complex structures are discussed. All main dynamic effects are substantiated by experimental observations at lab scale as well as in real applications.