Quillen's higher K-groups, defined in 1971, paved the way for motivic cohomology of algebraic varieties. Their definition as homotopy groups of combinatorially constructed topological spaces initially seems abstract and inaccessible. In this talk, aimed at a general audience, we will present Quillen's definition along with a new alternative algebraic definition, the first one not involving homotopy groups. The new ingredients are a bit more complicated than the chain complexes of homological algebra -- I will try to motivate, heuristically, why they have something to do with higher K-groups.