Organ transplants and the immune system

Have you heard about organ transplant patients having to take anti-rejection drugs?  The people involved with transplanting organs work to match the available organs and the receiving patient’s blood and body tissue as closely as possible. However, often the recipient must take anti-rejection drugs to quiet his body’s immune system and prevent it attacking (rejecting) the organ.   In this situation, the body’s normal response of ‘self’ vs. ‘non-self’ would result in an attack on the new organ because the body recognizes it’s a foreign tissue. The anti-rejection drugs would suppress this normal response.