Partial deduction in the Lloyd±Shepherdson framework cannot achieve certain optimisations which are possible by unfold/fold transformations. We introduce conjunctive partial deduction, an extension of partial deduction accommodating such optimisations, e.g., tupling and deforestation. We ®rst present a framework for conjunctive partial deduction, extending the Lloyd±Shepherdson framework by considering conjunctions of atoms (instead of individual atoms) for specialisation and renaming. Correctness results are given for the framework with respect to computed answer semantics, least Herbrand model semantics, and ®nite failure semantics. Maintaining the well-known distinction between local and global control, we describe a basic algorithm for conjunctive partial deduction, and re®ne it into a concrete algorithm for which we prove termination. The problem of ®nding suitable renamings which remove redundant arguments turns out to be important, so we give an independent technique for this. A fully automatic implementation has been undertaken, which always terminates. Dierences between the abstract semantics and Prolog's left-to-right execution motivate deviations from the abstract technique in the actual implementation, which we discuss. The implementation has been tested on an extensive set of benchmarks which demonstrate that conjunctive partial