  We present a class of nonconforming virtual element methods for general
fourth order partial differential equations in two dimensions. We develop a
generic approach for constructing the necessary projection operators and
virtual element spaces. Optimal error estimates in the energy norm are provided
for general linear fourth order problems with varying coefficients. We also
discuss fourth order perturbation problems and present a novel nonconforming
scheme which is uniformly convergent with respect to the perturbation parameter
without requiring an enlargement of the space. Numerical tests are carried out
to verify the theoretical results. We conclude with a brief discussion on how
our approach can easily be applied to nonlinear fourth order problems.
