Abstarct: The study of quantum systems with an explicit time-dependence has received great attention due to application in various fields of physics. Many of interesting quantum effects, including geometric Berry phase, molecular Bohm effect, dynamical locality of particles and Hall effect are related to this concept. More generally speaking, the time-dependent fields are also present in quantum chemistry, quantum optics and quantum field theory. In other words, the time-dependent models enable us to simulate the modern and applied concepts in these branches. These models become particularly more important when we consider the experimental limitations. In this thesis, we first thoroughly investigate the nonrelativistic particles using Schrödinger equation via time-space translation and Lewis-Riesenfeld approaches. Next, we study the relativistic particles using Dirac and Klein-Gordon fields and analyze them in various states including constant electric field and moving potential. In addition, we investigate ultra-relativistic particles in electromagnetic fields and solve Heun and Mathieu equations in our calculations.