We studied the influence of water on the structural stability and transformations of MAPI and FAPI by anelastic and dielectric spectroscopies under various temperature and H2O partial pressure protocols. Before discussing the new results in terms of interstitial water in MAPI and FAPI, the literature is briefly reviewed, in search of other studies and evidences on interstitial water in hybrid halide perovskites. In hydrated MAPI, the elastic anomaly between the cubic α and tetragonal β phases may be depressed by more than 50%, demonstrating that there are H2O molecules dispersed in the perovskite lattice in interstitial form, that hinder the long range tilting of the PbI6 octahedra. Instead, in FAPI, interstitial water accelerates in both senses the reconstructive transformations between 3D α and 1D δ phases, which is useful during the crystallization of the α phase. On the other hand, the interstitial H2O molecules increase the effective size of the MA and FA cations to which are bonded, shifting the thermodynamic equilibrium from the compact perovskite structure to the open δ and hydrated phases of loosely bonded chains of PbI6 octahedra. For this reason, when fabricating devices based on hybrid metal-organic halide perovskites, it is important to reduce the content of interstitial water as much as possible before encapsulation.