I just read a nice piece, Suber's "A crash course in the mathematics of infinite sets". No "higher" math required, just sharp thinking. Suprising results, with simple, clear path to them. I believe this is mathematics at it's best.
As traditional museums try to show the beauty of paintings and sculptures, museums of mathematics also try to do the same with Mathematics. Here are two of them:
Erlebnisland Mathematik | Technische Sammlungen Dresden
MoMath | National Museum of Mathematics
There also big math festivals for the general public:
National Math Festival
Festival da ...