This doctoral thesis deals with bisimulations and bisimulation games for Verbrugge semantics of interpretability logic. First, bisimulations and finite bisimulations for Verbrugge semantics that have been used in the literature so far are considered. The properties they possess are proved and the problems that arise when using these notions are highlighted. Then, new notions of the bisimulations and finite bisimulations are given, which are called weak bisimulations and finite weak bisimulations (or w-bisimulations and n-w-bisimulations). It is proved that these new notions still have all the good properties of bisimulations and finite bisimulations, but that they also possess more good properties, i.e. that there are no problems that occurred with earlier definitions. In order to obtain a simpler technique for proving (n-)w-bisimilarity of two worlds in Verbrugge models, it is proved that (n-)w-bisimilarity of worlds is equivalent to the winning strategy of the defender in the corresponding games (called (n-)w-bisimulation games). In the next part of the thesis, new definitions related to the tools used in proving the characterization theorem are presented, namely standard translation and q-saturated unravelling. More precisely, the signature of the language of the two-sorted first-order logic is defined, into whose formulas the standard translation will map the formulas of the interpretability logic IL. Using the properties of q-saturated unravelling and the selection method, the finite model property of interpretability logic IL is proved. Finally, using previously introduced concepts and proved results, the main result of this paper is proved. It is an analogue of van Benthem’s characterization theorem for the logic of interpretability IL with respect to Verbrugge’s semantics, which shows that the logic of interpretability with Verbrugge’s semantics precisely corresponds to a fragment of the two-sorted first-order logic that is invariant to w-bisimulations.