The unusual electronic bandstructure of graphene is shown to give it several novel properties beyond the much-studied transport behavior. The electron density response of a graphene sheet is studied here in some detail. It is thereby shown that the attractive forces between graphene and distant objects can be calculated analytically, and exhibit unusual power laws as a function of their separation D. A brief review is also given of recent progress with high-level numerical many-electron calculations involving graphene interactions near the equilibrium binding distance D_0.