  In this paper we initiate the investigation of Virtual Elements with curved
faces. We consider the case of a fixed curved boundary in two dimensions, as it
happens in the approximation of problems posed on a curved domain or with a
curved interface. While an approximation of the domain with polygons leads, for
degree of accuracy $k \geq 2$, to a sub-optimal rate of convergence, we show
(both theoretically and numerically) that the proposed curved VEM lead to an
optimal rate of convergence.
