Two new approximations for multivariate normal probabilities for rectangular regions, based on conditional expectations and regression with binary variables, are proposed. One is a second-order approximation that is much more accurate but also more numerically time-consuming than the first-order approximation. A third approximation, based on the moment-generating function of a truncated multivariate normal distribution, is proposed for orthant probabilities only. Its accuracy is between the first- and second-order approximations when the dimension is less than seven and the correlations are not large. All of the approximations get worse as correlations get larger. These new approximations offer substantial improvements on previous approximations. They also compare favorably with the methods of Genz for numerical evaluation of the multivariate normal integral. The approximation methods should be especially useful within a quasi-Newton routine for parameter estimation in discrete models that involve the multivariate normal distribution.