The main goal of my research work is on studying of Generalized Willmore surfaces and flows, by combining methods from pure mathematics (Differential Geometry and PDE) and applied mathematics (finite element method, numerical analysis, computations and modeling of surfaces and flows). I am interested in studying of modeling of surfaces and flows both from a theoretical and from a numerical point of view. My Master thesis was directed by Dr. Magdalena Toda and is entitled “Constant Mean Curvature Surfaces of Revolution versus Willmore Surfaces of Revolution: A Comparative Study with Physical Applications”. This thesis studies some special types of surfaces of revolution and their real world applications. The main two cases hereby considered are the constant mean curvature surfaces of revolution (also called Delaunay surfaces) and Willmores surfaces of revolution, respectively. We first presented some geometric results on Delaunay CMC surfaces which correspond to certain classes of ordinary differential equations and then performed the original construction of Delaunay surfaces, based on roulettes of conics, after which we characterize these geometric objects as solutions to specific ODEs. As a physical application we pre- sented a few physical models of Delaunay surfaces arising as liquid bridges between two vertical walls - which are proved to be unduloidal surfaces, by using Calculus of Variations. We numerically computed the profile curves of these surfaces and provided some numerical models for them. By contrast, we studied Willmore surfaces as minimizers of the Willmore energy (or bending energy).