This is a nice tool for spreading mathematical culture and ideas among mathematicians. It starts with the notion of current (after de Rham) and ends with the use in the subject of top level research and new extremely powerful methods of the author (around 1991). In the middle it is shown how to use the notion of positive current to define and work (via the integration current of the fundamental class of a subvariety) in Intersection Theory (key words: Lelong numbers and multiplicities). The tools come from Analysis and bring (and often solve) with them several interesting problems which cannot be formulated in a purely algebraic way inside Algebraic Geometry. But these methods are very, very strong competitors even on natural very important algebraic problems. Of course, in this paper most of the proofs are omitted, but ideas and difficulties are not skipped. It is pleasant reading and even specialists in not too far fields can find here some ideas/tools useful for their job; everybody can find some recent deep idea (mostly from Demailly brain).