  In this paper we introduce the notion of an assembler, which formally encodes
"cutting and pasting" data. An assembler has an associated $K$-theory spectrum,
in which $\pi_0$ is the free abelian group of objects of the assembler modulo
the cutting and pasting relations, and in which the higher homotopy groups
encode further geometric invariants. The goal of this paper is to prove
structural theorems about this $K$-theory spectrum, including analogs of
Quillen's localization and d\'evissage theorems. We demonstrate the uses of
these theorems by analysing the assembler associated to the Grothendieck ring
of varieties and the assembler associated to scissors congruence groups of
polytopes.
