This talk is concerned with self-intersections (or multiple points) of multivariate Gaussian random fields. We show that multiple points do not exist in critical dimensions. The main result is applicable to fractional Brownian sheets, solutions to the stochastic heat and wave equation with spatially homogeneous Gaussian noise that is white in time and colored in space. This talk is based on a joint paper with R. C. Dalang, Cheuk Yin