As the title suggests, the primary concern of this study is with problems arising from a very widely used notion in the recent literature of linguistics and philosophy, the notion of implicature. As this concept was introduced and developed by the philosopher H. P. Grice, the main part of the study will understandably be centred around his work. Grice distinguished between two main types of implicature, the conventional and the conversational. In the first part we will be concerned with, what Grice called, conventional implicature, and in particular with the linguistic items generating it, as described in his work. Thus the aim of this part of the study will be to investigate the nature of conventional implicata, and to ask whether they can be justifiably claimed to be nonconsequential for truth-evaluation and invariable, as Grice argues. Grice's account in this respect will be found to be partly implausible, as regards his treatment of 'therefore', and partly inadequate, as it fails to take into account the wide ranging function of 'but' - his paradigm of conventional implicature - but treats its variable meaning aspects as invariable, conventional implicature. In view of the intriguing linguistic behaviour of 'but', the main contributions to this topic in the literature will be reviewed. In the second part of the study our primary aim will be to consider in detail linguistic phenomena that come under the rubric of conversational implicature in the literature - with an emphasis on Grice's examples - with a view to detecting common characteristics that can be taken as the parameters along which these phenomena can be defined as a homogeneous class. It will be concluded that they cannot. More stringent criteria will be proposed for membership in a narrowly defined class of conversational implicature. Two classes of background knowledge and assumptions will be described and shown to bear significantly on language production and understanding and, in particular, on the production and understanding of linguistic facts that have been called conversational implicatures. It will be concluded that the term 'conversational implicature' has been misused and abused. The view taken here will be that background knowledge schemes must be taken into account and represented in a language theory, though the difficulties facing such an enterprise are well understood and acknowledged. However, the overall conclusion will be that Grice's proposal effects a cut and dried demarcation between a neat but narrowly defined truth-functional semantics, on the one hand, and an unexplicated pragmatics, on the other, that would, however, include the most intriguing aspects of language use. , This view of language is not very revealing and, hence, uninteresting and unappealing.