For a vector bundle E on a projective variety X, the condition of having no intermediate cohomology- namely, being "arithmetically Cohen-Macaulay"- imposes very strong restrictions on E. For instance, thanks to Horrocks' theorem, we know that any aCM bundle on the projective space should totally split.

Ulrich proved that for aCM bundles there exists a bound on the dimension of their space of global sections. Therefore, we can even strengthen the requirements and pay attention to those aCM bundles attaining this bound (i.e., Ulrich bundles).

Despite all of these constraints, it was conjecture by Eisenbud and Schreyer that any projective variety supports an Ulrich bundle. In this talk, I intend to report on the history of this problem as well as to explain the contributions that, jointly with Rosa Maria Miró-Roig, we have obtained.