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. Standard techniques in the context of parametric excitation are outlined like the Floquet theory, the method of averaging, the method of multiple scales and the complexification method which enable the (semi-) analytical description of stability boundaries in the system parameter space. Armed with these tools, analytical and numerical investigations of both multibody and continuous mechanical/mechatronic systems under parametric and external periodic excitations are discussed. Analytical studies are employed to detect and analyze stationary and non-stationary resonances. Numerical and experimental investigations are carried out for analyzing regular, chaotic, bifurcation and synchronization phenomena. The not well-known coexistence phenomenon and disappearance of resonance tongues is discussed in detail (Ince's generalization of Mathieu's equation) and the effects of quasi-periodic forcing and of time delay on the system dynamics are outlined. One major point for real applications is the ability of a specific procedure to analyze (non-)linear, time-periodic systems consisting of large number of degrees of freedom. The classic Floquet method cannot be applied to this kind of system due to the high sensitivity of the results on numerical round-off errors. The symbolic calculation of the Lyapunov transformation matrix based on the Chebyshev polynomials is introduced which enables a transformation of the original time-periodic problem to a time-invariant problem and down- scale the system to the relevant variables. A symbolic control technique for Floquet multiplier placement for linear time-periodic systems and bifurcation control and feedback linearization techniques for nonlinear time-periodic systems are presented. Reduction procedures of PDEs to ODEs are discussed with an emphasis on the reliability and validity of the obtained results. Advantages of construction of charts of vibration types and application of the Morlet wavelets and Lyapunov exponents are illustrated. Another branch of this course covers linear and nonlinear effects of high-frequency (HF) time-periodic excitation which lead to the main non-trivial effects like stiffening, biasing and smoothening and highlights their practical importance. Generalized results for nonlinear systems with rapid time-harmonic excitation are derived and extended to nonlinear discrete mechanical systems and linear continuous systems. The connection to practical examples is given by using HF excitation to quench friction-induced vibrations and resolving the famous Chelomei’s pendulum. The dynamic behaviour of discontinuous time-periodic systems covering different types of discontinuities like mechanical slip, stop and friction is discussed. Asymptotic methods are generalized for non-smooth and discontinuous systems. Friction can be the energy source for exciting vibrations but can also be used for quenching vibrations excited by periodic forces. Methods for analysis of oscillations in strongly damped systems are derived and these tools are then exploited to design a sequential friction- spring for quenching resonance or understand the resonance in systems with inelastic collisions. The positive effect of a parametric combination resonance, the so-called parametric anti-resonance is discussed analytically and numerically for simple 2DOF systems as well as MDOF systems. The conditions under which a parametric excitation becomes a parametric anti-resonance are derived in detail. These conditions allow interpreting the attenuation effect as a modal interaction due to time-periodicity. The major findings are verified experimentally and future application fields are outlined. This course offers a condensed view on time-periodic systems, their analytical and numerical analysis and avoidance or exploitation of parametric resonances. The underlying theory is explained and applied to simple and real-life engineering examples.