In recent years, functional methods have become central to the study of theoretical and applied mathematical problems. They not only provide generality but can also unify results and techniques and may lead to better results than those obtained by classical methods. This Research Note uses functional methods to explore a broad range of elliptic, parabolic, and hyperbolic boundary value problems and various classes of abstract differential and integral equations. The authors discuss a wealth of interesting applications relating to physical phenomena and offer a survey of the primary techniques used in the area, including their own important contributions to the field.