  Until recently, a quantum instrument was defined to be a completely positive
operation-valued measure from the set of states on a Hilbert space to itself.
In the last few years, this definition has been generalized to such measures
between sets of states from different Hilbert spaces called the input and
output Hilbert spaces. This article presents a theory of such instruments.Ways
that instruments can be combined such as convex combinations, post-processing,
sequential products, tensor products and conditioning are studied. We also
consider marginal, reduced instruments and how these are used to define
coexistence (compatibility) of instruments. Finally, we present a brief
introduction to quantum measurement models where the generalization of
instruments is essential. Many of the concepts of the theory are illustrated by
examples. In particular, we discuss Holevo and Kraus instruments.
