The Virtual Element method is a very recent variant of Finite Element Methods, appeared on the scene a couple of years ago. The method could be seen as a combination of Finite Element Methods and Mimetic Finite Differences. In particular it allows, at the same time, the use of very general decompositions of the computational domain (in almost arbitrary polygons or polyhedra), and the use of the (more elegant and more clarifying) Galerkin framework for the analysis and the error estimates. The talk, addressed to a rather general audience of mathematicians, will describe first the general idea of the method on a very simple problem like Poisson problem in 2 dimensions, and then give some very quick hints on various generalizations, including 3D problems, Stokes problem, Kirchhoff plates, variable coefficients, mixed formulations, etc. Much more details could be given, after the lecture, to the interested people. Several papers (both on the basic principles and on further extensions) can be found (and downloaded) on the web page of the speaker: http://www.imati.cnr.it/brezzi/rec_pubbl.html (the newest are at the end).