This paper discusses distributionally robust geometric programs with individual and joint chance constraints. Seven groups of uncertainty sets are considered: uncertainty sets with first two order moments information, uncertainty sets constrained by the Kullback-Leibler divergence distance with a normal reference distribution or a discrete reference distribution, uncertainty sets with known first moments or known first two order moments information and nonnegative support, and the joint uncertainty sets for the product of random variables. For the seven groups of uncertainty sets, we find tractable reformulations of the distributionally robust geometric programs with both individual and joint chance constraints. Finally, numerical tests are carried out on a shape optimization problem.