Let {X-n, n greater than or equal to 1} be a sequence of standard Gaussian random vectors in IRd, d greater than or equal to 2. In this paper we derive lower and upper bounds for the tail probability P{X-n > t(n)} with t(n) is an element of IRd some threshold. We improve for instance bounds on Mills ratio obtained by Savage (1962, J. Res. Nat. Bur. Standards Sect. B, 66, 9396). Furthermore, we prove exact asymptotics under fairly general conditions on both X-n and t(n), as \\t(n)\\ --> infinity where the correlation matrix Sigma(n) of X-n may also depend on n.