The Tukey family of g-and-h distributions is often used to model univariate real-world data. There is a paucity of research demonstrating appropriate multivariate data generation using the g-and-h family of distributions with specified correlations. Therefore, the methodology and algorithms are presented to extend the g-and-h family from univariate to multivariate data generation. An example is provided along with a Monte Carlo simulation demonstrating the methodology. In addition, algorithms written in Mathematica 7.0 are available from the authors for implementing the procedure.