We survey old and new approaches to the study of symbolic powers of ideals.
Our focus is on the symbolic Rees algebra of an ideal, viewed both as a tool to
investigate its symbolic powers and as a source of challenging problems in its
own right. We provide an invitation to this area of investigation by stating
several open questions.Comment: The changes in v2 are small, and mostly consist of typo correction

Grifo, Eloísa