  The Virtual Element Method (VEM) is a very effective framework to design
numerical approximations with high global regularity to the solutions of
elliptic partial differential equations. In this paper, we review the
construction of such approximations for an elliptic problem of order $p_1$
using conforming, finite dimensional subspaces of $ H^{p_2}(\Omega)$, where
$p_1$ and $p_2$ are two integer numbers such that $p_2 \geq p_1 \geq 1$ and
$\Omega\in R^2$ is the computational domain. An abstract convergence result is
presented in a suitably defined energy norm. The space formulation and major
aspects such as the choice and unisolvence of the degrees of freedom are
discussed, also providing specific examples corresponding to various practical
cases of high global regularity. Finally, the construction of the "enhanced"
formulation of the virtual element spaces is also discussed in details with a
proof that the dimension of the "regular" and "enhanced" spaces is the same and
that the virtual element functions in both spaces can be described by the same
choice of the degrees of freedom.
