Cutting and pasting in algebraic geometry

Given some class of "geometric spaces", we can make a ring as follows.
(i) (additive structure) When U is an open subset of such a space X,
[X] = [U] + [(X \ U)];
(ii) (multiplicative structure) [X x Y] = [X][Y].
In the algebraic setting, this ring (the "Grothendieck ring of varieties") contains surprising structure, connecting geometry to arithmetic and topology. I will discuss some remarkable statements about this ring (both known and conjectural), and present new statements (again, both known and conjectural). A motivating example will be polynomials in one variable. (This talk is intended for a broad audience.) This is joint work with Melanie Matchett Wood.