With the emerging of metal-organic frameworks over the past few years, the quest for developing zeolite-like metal-organic frameworks (ZMOFs) has attracted much interest recently. Analogue to cation-exchanged zeolite structures, the presence of nonframework ions in metal-organic frameworks can increase the strength of interactions between the host and the guest molecules and in turn enhances the storage, separation and ion-exchange capability of the framework. A wealth of studies on zeolitic materials have showed that the static and dynamic properties of the nonframework ions are crucial to understanding the fundamental chemistry of these materials and also in tailoring their properties. In this work, using the atomistic simulations we explore the distribution and mobility of the nonframework ions in ZMOFs with both anionic and cationic frameworks. The nonframework ions are found to be essentially immobile and show an oscillating motion around their favorable sites. Hydrogen adsorption in the ZMOFs is also investigated.