In this paper we propose a new semantics, based on the notion of a "contextual model", that makes it possible to express and compare — within a unique formal framework — different views on the roles of various notions of context in knowledge ascriptions. We use it to provide a logical analysis of such positions as skeptical and moderate invariantism, contextualism, and subject-sensitive invariantism. A dynamic formalism is also proposed that offers new insights into a classical skeptical puzzle.