  The objective of this contribution is to develop a convergence analysis for
SUPG-stabilized Virtual Element Methods in diffusion-convection problems that
is robust also in the convection dominated regime. For the original method
introduced in [Benedetto et al, CMAME 2016] we are able to show an "almost
uniform" error bound (in the sense that the unique term that depends in an
unfavorable way on the parameters is damped by a higher order mesh-size
multiplicative factor). We also introduce a novel discretization of the
convection term that allows us to develop error estimates that are fully robust
in the convection dominated cases. We finally present some numerical result.
