It was once believed that identical (monozygotic or MZ) twins occurred at random. There is now some evidence to suggest that MZ twins may run in families (Machin 2009 – Familial monozygotic twinning: A report of seven pedigrees). In general, it is fraternal (dizygotic or DZ) twins that run in families. It is believed that there are genetic factors that can cause hyperovulation, which is the release of multiple eggs during ovulation. DZ twins occur when two eggs are released during the same menstrual cycle and are fertilized by two different sperm cells. The genetic factors that may cause hyperovulation have not been identified. There is no genetic testing available for them.
Sometimes twins “skip a generation” in some families. This is because genetic factors that may cause hyperovulation resulting in DZ twins can only be observed in women. But these genetic factors may be inherited by men and passed on to their daughters. Their daughters would then have an increased chance of having DZ twins.
So, if you are a female DZ twin and there are other DZ twins in your family, you may have an increased chance of having DZ twins. If you are a male DZ twin and there are other DZ twins in your family, you do not have an increased chance of having DZ twin children. But if you have daughters, you may have an increased chance of having DZ twin grandchildren!
The above discussion does not apply to DZ twins who were conceived by the use of fertility treatments.