In the last thirty years, there has been a growing interest in geometric energies, as for example the Perimeter functional or the Willmore functional. Some relevant problems regard the study of the associated geometric flows and the existence of critical points of suitable types. These questions are usually difficult to answer and sometimes it is useful to approximate the original functionals by simpler ones. The first part of the seminar will be very introductory, we will recall the definition of Gamma-convergence and its main consequences to state precisely what we mean by approximation. Moreover, we will briefly review the celebrated approximation of the Perimeter functional by means of the Modica-Mortola functionals. In the second part we shall discuss a conjecture of De Giorgi, dating back to 1991, where he proposed a possible approximation of the Willmore functional based on the first variation of the Modica-Mortola functionals. We will outline some of the most relevant contributions originated from this conjecture over time. Finally, we present a recent result obtained in collaboration with G. Bellettini and N. Picenni where we give a negative answer to the original conjecture.