This is notably the origin of the term "grandfather clause", although its current meaning is subtly different from the original sense.

The 15th Amendment made it illegal for states to deny the right to vote directly on the basis of race, so instead they imposed new poll taxes and "literacy tests" (which were not designed as true tests, but rather as instruments to allow poll workers to arbitrarily "pass" and "fail" would-be voters), while exempting those whose grandfathers were eligible to vote before a specific date.