This article provides a general expression for infinitely divisible multivariate gamma distributions defined by their Laplace transforms, as well as the conditional Laplace transform of infinitely divisible multivariate gamma distributions. We give algorithms for simulating infinitely divisible gamma distributions and infinitely divisible multifactor gamma distributions in dimension 2, 3, 4 and for all dimensions greater than 2 in the Markovian case. We give examples of simulations in dimension 2, 3, 4 and in dimension 5 in the Markovian case.