The dynamics of spiral galaxies is a gold-mine of challenging problems for the astrophysicist and the applied mathematician. In Astrophysics, we may ask how these island-universes formed, evolved, and reached their current structure and thus address the problems of the dynamics of the interstellar medium, of star formation, of stellar dynamics, and of the presence and role of dark matter halos. These fundamental issues have attracted the interest of the scientific community for almost a century. The models conceived and developed to sharpen and to quantify our understanding of the basic dynamical processes at work in this context require a semi empirical approach and general tools that are characteristic of Applied Mathematics. In particular, the beautiful morphology of spiral galaxies poses a number of interesting questions, most of which have found a coherent answer in the framework of the Density Wave Theory. I will give a synthetic description of the main concepts and of the main achievements of the theory, as developed in the course of three decades in the second part of last century. The theory has had a major impact on Astrophysics and has been the inspiration for a number of important theoretical and observational investigations. The relatively recent advent of near-infrared observations (especially those in the K-band, probing the underlying stellar component in galaxies) has confirmed that indeed large-scale spiral arms are associated with a smooth, sinusoidal density perturbation in the stellar disk, that grand design is very frequent and generally two-armed, and that multiple-armed spiral structure is mostly associated with the gaseous interstellar medium. I will also briefly outline other interesting topics in the dynamics of spiral galaxies, where progress may take place as a result of a fruitful exchange between Astrophysics and Applied Mathematics.