We propose new methods for solving the variational inequality problem where the underlying function F is monotone. These methods map be viewed as projection-type methods in which the projection direction is modified by a strongly monotone mapping of the form I - alpha F or, if F is affine with underlying matrix M of the form I + alpha M(T), with alpha is an element of (0, infinity). We show that these methods are globally convergent, and if in addition a certain error bound based on the natural residual holds locally, the convergence is linear. Computational experience with the new methods is also reported.