Elliptic operators of fractional order were popularized, mainly thanks to Luis Caffarelli, during the early 2000's. Suddenly, we learnt that every classical PDE had a fractional counterpart (or even more than one in some cases!). Also, fractional versions of most important techniques and results in PDE were developed. In this context, the invention in the late 2000's of fractional minimal surfaces may not seem a very striking milestone. Over the years, however, the interest and depth of these new surfaces is becoming unquestionable, to the point that they may be a fundamental tool in order to better understand certain (famously delicate) questions on classical minimal surfaces, such as Yau's conjecture. In the talk I will describe some very recent works that, I hope, may help to convince a fraction of the remaining skeptics about the beauty and usefulness of nonlocal minimal surfaces.
