Ellipsis is a pervasive phenomenon across the world’s languages, and it is easy to see why: it allows speakers to omit certain parts of their utterances while nonetheless conveying their full meaning, which contributes to making linguistic communication highly efficient. There is broad consensus that elliptical utterances depend on the context in some way, but the nature of this dependency remains controversial. Broadly speaking, theories of ellipsis fall into one of two camps that make fundamentally distinct architectural assumptions about the mechanisms that enable ellipsis: IDENTITY theories, which posit that material can be elided only if it is identical to a linguistic antecedent; and referential theories, which assume that ellipsis is enabled by the same underlying mechanism that governs other forms of discourse reference. This thesis draws on evidence from a series of experiments that investigate two prominent types of ellipsis: VP-ellipsis and sluicing. It focuses on cases of mismatch between the elided material and its antecedent in which the meaning of the ellipsis clause is not reducible to the linguistic context and must instead be inferred. Taken together, this investigation makes three main contributions. First, it raises novel challenges for IDENTITY theories by demonstrating that ellipsis can be felicitous even in the face of extreme mismatch. Secondly, it provides new evidence in support of referential theories by showcasing the possibility of inferential ellipsis resolution—a hallmark property of discourse reference. Finally, and most importantly, it contributes new adequacy criteria for linguistic theories aimed at explaining the nature of the linguistic and non-linguistic context and how it interfaces with context-dependent linguistic devices.