Heat and electrical contact resistances exist in composite superconductors between the normal metal and the superconductor in most cases. Frequently by some reasons these resistances are high enough. It is shown in this paper that in composite superconductors with noticeable contact resistance the existence of the resistive regions of finite size is possible. Such regions (resistive domains) are stable in the regime of fixed current and the temperature and the current density are non-uniform in the cross-section of the composite superconductor in the vicinity of the resistive domain. We find the I-V characteristic of a sample with a resistive domain and the minimum domain existence current ir. It is shown that the current iris much less than the minimum normal zone propagation current ip. We investigate the process of the resistive domain formation due to the heat pulse. Note that the resistive domains are unstable in the regime of fixed current in uniform composite superconductors without the contact resistance and are stable in non-uniform ones. The existence of the stable resistive domains largely regulates the process of the superconductivity destruction and the following recovery of the superconductivity in the presence of the transport current. Moreover the resistive domains existence is responsible for some hysteresis phenomena accompanying the superconductivity destruction and recovery. Some previous results were published in the short note.