  We present and analyze a Virtual Element Method (VEM) of arbitrary polynomial
order $k\in\mathbb{N}$ for the Laplace-Beltrami equation on a surface in
$\mathbb{R}^3$. The method combines the Surface Finite Element Method (SFEM)
[Dziuk, Elliott, \emph{Finite element methods for surface PDEs}, 2013] and the
recent VEM [Beirao da Veiga et al, \emph{Basic principles of Virtual Element
Methods}, 2013] in order to handle arbitrary polygonal and/or nonconforming
meshes. We account for the error arising from the geometry approximation and
extend to surfaces the error estimates for the interpolation and projection in
the virtual element function space. In the case $k=1$ of linear Virtual
Elements, we prove an optimal $H^1$ error estimate for the numerical method.
The presented method has the capability of handling the typically nonconforming
meshes that arise when two ore more meshes are pasted along a straight line.
Numerical experiments are provided to confirm the convergence result and to
show an application of mesh pasting.
