With growing water scarcity in many parts of the world and projections that indicate the need to increase agricultural production and, concurrently, agricultural water use, it is increasingly advocated to focus efforts on improving agricultural water productivity and efficiency - and thus achieve more crop per drop. Many international organizations concerned with water management are also promoting these efforts, and significant public and private investments are being made in both developed and developing countries. Yet some serious problems are associated with this approach. They include conceptual issues, the methods used for measuring agricultural water productivity and efficiency, and the application of these concepts and methods in different contexts - all of which influence the choice of interventions and the evaluation of their implementation. The report aims to shed further light on these issues: first, by clarifying some of the underlying concepts in the discussion of agricultural water productivity and efficiency; second, by reviewing and analyzing the available methods for assessing water productivity and efficiency, including single-factor productivity measures, total factor productivity indices, frontier methods, and deductive methods; and, third, by discussing their application and relevance in different contexts. As a background for this analysis, the report highlights the central role of water use in irrigated agriculture and its link with increasing water scarcity. An underlying framework of the analysis is the view of the water economy transitioning from an expansionary to a mature phase. The report further develops this framework to reflect water management issues in irrigated agriculture. The framework is then applied to make the case that, with increasing water scarcity, the ongoing efforts for improving agricultural water productivity and efficiency need to move beyond crop per drop approaches, because they are in many circumstances an insufficient and sometimes counterproductive attempt to adapt agricultural water management to a maturing water economy.