An apparatus comprising a high flow rate respiratory gas source and a nasal cannula assembly connected to the respiratory gas source for supplying a respiratory gas to a patient, the nasal cannula assembly comprising a nasal cannula having spaced apart cannula arms and a pair of supply lines which each have a head at one end thereof with a discharge opening therein for discharging a respiratory gas, and the opposite end of each of the pair of supply lines being connected to an auxiliary respiratory gas supply line, and a remote end of each of the auxiliary respiratory gas supply line is connected with a respiratory gas source configured to supply a constant flow rate of a respiratory gas to the patient; a central bridge member of the nasal cannula being located adjacent the heads for integrally connecting to the supply lines to one another; and each head being sized to be snugly received and retained within one of the nasal cavities of the patient while forming leakage passages, between a portion of inwardly facing nasal cavity skin of a patient and a portion of an exterior surface of the head, which facilitate exhausting of exhaled gases and exhausting of excess respiratory gas supplied to the patient and the leakage passages continuously allowing room air to flow inwardly therethrough in an event that additional air, for breathing by the patient in excess to the constant flow rate of the respiratory gas, is required during inhalation; each leakage passage is formed as an elongate trough in an exterior surface of the head; wherein the bridge member defines an internal flow passage therein which provides an interconnecting gas flow path between the spaced apart cannula arms and generally operates to equalize the flow of gas from the pair of supply lines to the pair of heads and provides an alternate flow in the event that one of the heads becomes partially or completely obstructed or restricted during use; characterized in that: the supply lines, the heads and the discharge openings are configured to supply the respiratory gas, from the respiratory gas source, at a flow rate of about 120 liters per minute and at a pressure of between 4 and 20 cm of H 2 0; and the nasal cannula assembly forming an open system such that when the respiratory gas is introduced into the nostrils of the patient's nose, a portion of the respiratory gas, along with a portion of the exhaled gases from the patient, being allowed to leak out through the leakage passages formed in the exterior surface of the heads, whereby the flow resistance through the nasal cannula is determined and controlled by the leakage passages formed between the exterior surface of the heads of the nasal cannula and the mating interior skin of the patient's nasal passage, the heads being enlarged to facilitate secure retention of the nasal cannula within a nostril of a patient while the leakage passages prevent a fluid tight seal from being formed between the exterior surface of the enlarged head of the nasal cannula and the inwardly facing skin tissue of the patient's nostril, the dimensions of the leakage passages and the gas flow rate being selected so as to maintain a sufficient positive back pressure within the patient's breathing passages so that the patient's breathing passages always remain adequately inflated during the entire breathing cycle and, in particular, during exhalation. the supply lines, the heads and the discharge openings are configured to supply the respiratory gas, from the respiratory gas source, at a flow rate of about 120 liters per minute and at a pressure of between 4 and 20 cm of H 2 0; and the nasal cannula assembly forming an open system such that when the respiratory gas is introduced into the nostrils of the patient's nose, a portion of the respiratory gas, along with a portion of the exhaled gases from the patient, being allowed to leak out through the leakage passages formed in the exterior surface of the heads, whereby the flow resistance through the nasal cannula is determined and controlled by the leakage passages formed between the exterior surface of the heads of the nasal cannula and the mating interior skin of the patient's nasal passage, the heads being enlarged to facilitate secure retention of the nasal cannula within a nostril of a patient while the leakage passages prevent a fluid tight seal from being formed between the exterior surface of the enlarged head of the nasal cannula and the inwardly facing skin tissue of the patient's nostril, the dimensions of the leakage passages and the gas flow rate being selected so as to maintain a sufficient positive back pressure within the patient's breathing passages so that the patient's breathing passages always remain adequately inflated during the entire breathing cycle and, in particular, during exhalation.