It depends a lot on the degree and type of importance. If your dad is Mick Jagger, the last name will probably suffice. Less facetiously, there are still Roose

It depends a lot on the degree and type of importance. If your dad is Mick Jagger, the last name will probably suffice. Less facetiously, there are still Roosevelts and Cabbotts and Coolidges out there and those names do carry weight in some places. I imagine there are some names which would go pretty far at Oxford too. If your parents are more self-made (or the last name doesn't reflect the lineage), the personal statement can absolutely cover this. "Growing up in the shadow of my family" is partly about personal development, but partly a way to say "check out my family". Honestly, though? In the US, this gets handled via side channels. Our applications have a space for 'legacy status', where you list past attendees in your family. Anything counts, but rich and famous alumni are especially good. And if your family is important enough, they'll probably have talked to the school completely independent of your application. Likely, they'll have started years before you applied. I know several people from top US colleges who have a story about meeting some strangely unqualified classmate, and then discovering that they shared a name with some building on campus. At the very top end, there's certainly room to pave the way with donations and personal friendships.