The great beauty of VEM

In this paper I review the main features of the (newborn) Virtual Element Method, and of its application to the approximation of boundary value problems for Partial Differential Equations of particular relevance for applications. I will mostly concentrate on the definition of the Virtual Element spaces, that, roughly, consist of (vector valued) functions that are solution of (systems of) partial differential equations in each subdomain of a decomposition of the computational domain into polygons or polyhedra of quite general shape. Then I will give some hint on the use of these spaces for the discretization of some classical toy-problems like Heat conduction, Darcy flows, and Magnetostatic problems.