Multivariate extensions of the Gompertz distribution are plausible models for the study of several dependent populations with a Gompertz law of growth or multivariate survival data. In this paper we introduce a multivariate distribution with univariate marginal distributions of Gompertz-type form. The new distribution is expressed in closed form and shows symmetry in the component variables. We provide explicit expressions for the first moments which are functions of the Euler constant. Specifically we develop a trivariate Gompertz-type distribution and afterwards consider the multivariate case as an immediate extension of this. The problem of estimating the parameters of the new multivariate distribution is also discussed.