Inheritance law begins from its fundamental principles, and for this paper it is especially important the principle of escheat. Because of the fact that the testator died in the moment of his death, occurrence of inheritance takes place which means that the inheritance is passed on to his heirs and other appropriate legal effects take place. Our legal system provided the possibility of renunciation of the inheritance, meaning that a person doesn’t have to be an heir if it doesn’t want to. The inheritance is consisted of legal goods of the testator from one side, and of testator’s debts on the other. So, everything that was testator’s property in the moment of his death, assuming that is inheritable forms the inheritance. Heir’s responsibility, as a key question of this paper, is extremely important because in most of the cases, after the testator’s death there are debts left on regular basis, which are passed on to the heirs if they do not renounce of inheritance. One tries to protect the creditor by existing legislative solutions so that he can realize his claims despite the testator’s death. For this purpose, to ensure realizing his claims, the Law of Succession has given to the creditors the possibility of requesting separation of the inheritance. There is a difference in legal position regarding the responsibility of heirs, coheirs, necessary heirs and heirs according to delayed conditions with special terms within the each. However, there is a special problem when the testator has no heirs, i.e. when heirs renounce the inheritance and when such an inheritance without an heir becomes escheat. It is defined by law that this inheritance ex lege goes to the units of local self-government, districts and cities. In that case, they are on the position of the legal heir which also makes them responsible for the testator’s debts, without the possibility of renunciation of inheritance. This could have even jeopardized the existence of districts and cities, especially because of their modest budget until recently when last changes of the Inheritance Law brought somewhat satisfying solution regarding their responsibility, and providing the appropriate protection of cities and districts like their exception from execution, or the possibility of execution’s limitation of funds on accounts of local self-government’s units according to the Distraint law. It is important to comment on other forms of responsibilities, like lifelong support contract, contract of assignment and distribution of property during a lifetime, and on the responsibility of the heir from the tax debtor relationship with which it will be dealt with in the last chapter of this paper.