A methodology, to quantify the reliability of passive safety systems, proposed for use in advanced reactor design, is developed. Passive systems are identified as systems that do not need any external input or energy to operate and rely only upon natural physical laws (e.g. gravity, natural circulation, heat conduction, internally stored energy, etc.) and/or intelligent use of the energy inherently available in the system (e. g. chemical reaction, decay heat, etc.). The reliability of a passive system refers to the ability of the system to carry out the required function under the prevailing condition when required: the passive system may fail its mission, in addition to the classical mechanical failure of its components, for deviation from the expected behaviour, due to physical phenomena mainly related to thermalhydraulics or due to different boundary and initial conditions. The present research activity is finalized at the reliability estimation of passive B systems (i.e. implementing moving working fluids, cf. IAEA): the selected system is a loop operating in natural circulation including a heat source and a heat sink. This calls for the evaluation of the thermal-hydraulic unreliability that is the probability of failure of the natural circulation upon which the system operation relies, as essential contributor to the passive system reliability assessment.