I usually have a more or less formed template of conversation to talk with people about mathematics, It's importance, methods, history, etc. I've been for some time interested in newer fields of mathematics that helped to solve or solved older problems of mathematics but my level of mathematical culture is still too low and I probably don't have good examples in mind.
Although the question may seem an enthusiastic atempt to spread the word of how mathematics is to the laymen, I'm also interested in examples that aren't so accessible for two reasons:
I'm curious and perhaps I won't understand now but I have the future to get them.
It could be useful/interesting for other members of the community that are way more advanced than the laymen.