A good deal of what is communicated takes place "between the lines" of conventional meaning, relying on the speaker's and hearer's coordination of conversational expectations, and the juxtapositioning of these expectations with the conventional meaning of the utterance. Grice's work on conversational implicature has provided a useful framework for thinking about this important contribution to meaning by emphasizing the distinction between conventional and understood meanings, and sketching out a set of communicative principles through which understood meanings might be derived on the basis of conventional meanings. A critical feature of Grice's conception of conversational implicatures is the notion that they are calculable. Thus, a speaker who says p may implicate q: "PROVIDED THAT (1) he is to be presumed to be observing the conversational maxims, or at least the cooperative principle; (2) the supposition that he is aware that, or thinks that, q is required in order to make his saying p ... consistent with this presumption; (3) the speaker thinks (and would expect the hearer to think that the speaker thinks) that it is within the competence of the hearer to work out, or grasp intuitively, that (2) is required." (Grice, 1975, 49-50). However, actual language production and comprehension takes place at a frenetic pace, and involves heavy demands on the processing system under dramatic time pressures. Hence, the circumstances under which "Gricean" calculation may be implemented during real-time language processing are very much open to empirical discussion. Researchers disagree on conceptualizing the computational demands of implicature. For example, some have argued (e.g. Levinson, 2000) for a distinct class of generalized conversational implicatures that may be derived rapidly and automatically on the basis of general conversational principles and paradigmatic lexical relationships. Others (e.g. the Relevance theorists) have claimed that there is no principled or cognitive distinction between generalized implicature and particularized implicatures which clearly cannot be calculated without consideration of the particular conversational situation at hand. In this talk, I will review what is currently known about the processing costs of conversational implicature, suggesting that there is currently little evidence overall for a sharp dissociation in processing costs between generalized and particularized implicature. I will present some work from my own lab showing that conversational inferences can be derived "on the fly" with extreme rapidity during real-time language comprehension. Despite the speed of these inferences, they do not appear to be generated automatically without consideration of the particulars of the conversational situation; rather, they seem to depend upon the hearer's assessment of the extent to which the speaker is adhering to conversational principles. I will conclude by making some speculative hypotheses as to how the processing mechanism might yield such rapid, seemingly complex inferencing.