The Gaussian correlation conjecture states that for any two symmetric, convex sets in $n$-dimensional space and for any centered, Gaussian measure on that space, the measure of the intersection is greater than or equal to the product of the measures. In this paper we obtain several results which substantiate this conjecture. For example, in the standard Gaussian case, we show there is a positive constant, $c$ , such that the conjecture is true if the two sets are in the Euclidean ball of radius $c \sqrt{n}$. Further we show that if for every $n$ the conjecture is true when the sets are in the Euclidean ball of radius $\sqrt{n}$, then it is true in general. Our most concrete result is that the conjecture is true if the two sets are (arbitrary) centered ellipsoids.