This talk will be a gentle introduction to the Grothendieck ring of varieties (the "baby ring of motives") and its role in the search for computable invariants of varieties. This ring is important, mysterious, and in some ways pathological. I will describe various questions about the Grothendieck ring, motivated by arithmetic, topology, and geometry--in particular, an analogue of part of the Weil conjectures--as well as some of its pathologies. I'll answer some of these questions for curves and discuss how these answers manifest themes from arithmetic and topology. Time permitting, I'll discuss obstructions to the truth of related statements for surfaces