  We study the operational bivariant theory associated to the covariant theory
of Grothendieck groups of coherent sheaves, and prove that it has many
geometric properties analogous to those of operational Chow theory. This
operational K-theory agrees with Grothendieck groups of vector bundles on
smooth varieties, admits a natural map from the Grothendieck group of perfect
complexes on general varieties, satisfies descent for Chow envelopes, and is
A^1-homotopy invariant. Furthermore, we show that the operational K-theory of a
complete linear variety is dual to the Grothendieck group of coherent sheaves.
As an application, we show that the K-theory of perfect complexes on any
complete toric threefold surjects onto this group. Finally, we identify the
equivariant operational K-theory of an arbitrary toric variety with the ring of
integral piecewise exponential functions on the associated fan.
