  In this paper we study the subset of generalized quantum measurements on
finite dimensional systems known as local operations and classical
communication (LOCC). While LOCC emerges as the natural class of operations in
many important quantum information tasks, its mathematical structure is complex
and difficult to characterize. Here we provide a precise description of LOCC
and related operational classes in terms of quantum instruments. Our formalism
captures both finite round protocols as well as those that utilize an unbounded
number of communication rounds. While the set of LOCC is not topologically
closed, we show that finite round LOCC constitutes a compact subset of quantum
operations. Additionally we show the existence of an open ball around the
completely depolarizing map that consists entirely of LOCC implementable maps.
Finally, we demonstrate a two-qubit map whose action can be approached
arbitrarily close using LOCC, but nevertheless cannot be implemented perfectly.
