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This paper explores the semantics of the meta-notation used in the style of operational semantics introduced by Felleisen and Hieb. Specifically, it defines a formal system that gives precise meanings to the notions of contexts, decomposition, and plugging (recomposition) left implicit in most expositions. This semantics is not naturally algorithmic, so the paper also provides an algorithm and proves a correspondence with the declarative definition. The motivation for this investigation is PLT Redex, a domain-specific programming language designed to support Felleisen-Hieb-style semantics. This style of semantics is the de-facto standard in operational semantics and, as such, is widely used. Accordingly, our goal is that Redex programs should, as much as possible, look and behave like those semantics. Since Redex's first public release more than seven years ago, its precise interpretation of contexts has changed several times, as we repeatedly encountered reduction systems that did not behave according to their authors' intent. This paper describes the culimation of that experience. To the best of our knowledge, the semantics given here accommodates even the most complex uses of contexts available.