In this thesis, the electromagnetic wave propagation is studied in nonstationary-medium scenarios. The electromagnetic fields under material time-modulation are shown to conserve their momentum but not their energy. The mathematical foundations and analysis to treat wave propagation in time-Floquet media are given additionally to the related parametric amplification phenomena, which are mapped to the stability analysis of the corresponding hypergeometric equations. Assuming a time-variation of permittivity, permeability and conductivity the appropriate time-domain solutions are derived, based on an observation of the fields in the past. The formulation of a time-transitioning state matrix connects the unusual energy transitions of electromagnetic fields in time-varying media with the exceptional point theory, a theory strongly connected with parity-time symmetry. Consequently, the state-matrix approach of this thesis allows the analysis of the electromagnetic waves in terms of parity and time-reversal symmetries and signify parity-time symmetric wave-states without the presence of a spatially symmetric distribution of gain and loss, or any inhomogeneities and material periodicity. The parametric amplification phenomena of time-Floquet media and more precisely those that generate a Mathieu equation at the first momentum gap are theoretically studied and numerically compared with simulations using FDTD and connected with the parity-time scattering conventional characteristics. In the last part of this thesis, studies regarding resonant acoustic and electromagnetic systems are exhibited. The theoretical foundation to treat both acoustic and electromagnetic resonant phenomena is given based on the coupled mode theory and the appropriate Hilbert space. Two examples of interest are shown leveraging the time-dynamics of a temporal resonant system. The first example is related to the design of an artificial resonant acoustic lattice with the appropriate time-modulation leading to an effective zero index of refraction. The second example is related to resonant systems with temporal coupling and the possibility to induce nonreciprocal gain by leveraging the frequency conversion occurring in parametric systems. This thesis enriches the literature and the theoretical bases for dynamical wave systems and provides an insight on the broad capabilities of time-varying systems in electromagnetics, optics and acoustics. It may be used as a guidance to realize wave devices that amplify and actively filter wave signals for many future applications in lasing, sensing, signal amplifying, energy transferring and imaging.