Let X be a (N,d)-Gaussian random field. Under certain general conditions, we study the hitting probabilities of X and determine the Hausdorff dimension of the inverse image under X of a non-random Borel set. The class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion, the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space-time white noise and operator-scaling Gaussian random fields with stationary increments.