• calculus of variations
• geometric measure theory
• willmore conjecture
• willmore functional

This work is devoted to the study of the Willmore Functional.
In the first part of the thesis we recall some necessary prerequisites and we prove some main properties of the funtional, then we investigate the existence of the minimum under the most classical constraints and we study the main ideas in the proof of the Willmore Conjecture.
In the second part we consider minimization problems related with the Willmore Functional involving also the area of the surface or a volume related to it; we study existence of minimizers and we provide some examples.