My sons were born in 1966 and 1971.
This page (Encyclopedia) says the draft ended in 1973, but Carter re-instated registration requirements when Russia invaded Afghanistan in 1980.
Here's the excerpt about Carter:
End of the draft
The draft formally ended in 1973; young men no longer had to register with the Selective Service System. In 1980, the Soviet Union invaded Afghanistan, and American president Jimmy Carter (1924–; served 1977–81) reinstated the registration requirement. All males between the ages of eighteen and twenty-five who were born on or after January 1, 1960, were required to register. This law remained in place into the twenty-first century. Because prosecuting those who fail to register is cost prohibitive and counterproductive for the government, many young men fail to register on their eighteenth birthday or do not register at all.
It is a big deal. Not registering precludes you from working for the government or a government contractor.
Because prosecuting those who fail to register is cost prohibitive and counterproductive for the government, many young men fail to register on their eighteenth birthday or do not register at all.