SYMPOSIA C ereals, dominated by wheat, rice, and maize, provide approximately 50% of human food calories directly and con-siderably more indirectly via feed grains (Tweeten and Thomp-son, 2008). Over the last 20 yr, a period chosen to best estimate current rates of progress without infl uence of earlier periods, the linear rates of yield change for the world (Fig. 1) have been 25kgha –1 yr –1 (wheat), 38 kg ha –1 yr –1 (rice), and 80 kg ha –1 yr –1 (maize). With the exception of maize in some regions, there is no evidence for exponential growth in yield. In fact, relative rates of yield increase are declining and, expressed relative to predicted yield in 2007, are 0.9% yr –1 for wheat, 0.9% yr –1 for rice, and 1.6%yr –1 for maize. Even if these relative rates could be main-tained, various studies suggest they would not prevent real price rises for the three cereals, in the face of projected demand growth to 2050 (Tweeten and Thompson, 2008). Thus there is little doubt that the world needs to continue increasing cereal yields. In this paper we focus on factors determining current rates of yield progress in several key situations (or case studies) and consider ABSTRACT This paper reviews recent progress in wheat (Triticum aestivum L.), rice (Oryza sativa L.), and maize (Zea mays L.) yields resulting from substantial breeding efforts in mostly favorable environments and exam-ines its physiological basis. Breeding and improved agronomy lift potential yield (PY), namely yield with the best variety and management in the absence of man-ageable abiotic and biotic stresses, and PY increase is a key component of progress in farm yield (FY), the other component being closure of the PY to FY gap. Changes in PY and FY are reviewed for several key production regions, namely the United Kingdom and the Yaqui Valley of Mexico for wheat, Japan and Central Luzon in the Philippines for rice, and Iowa and briefl y sub-Saharan Africa for maize. The PY growth rates have fallen and are currently generally no more than 1% per annum and usually much less. The tra-jectory of FY with time often closely parallels PY, but, especially in developing countries, there remain large yield gaps. In at least one instance (maize in Iowa) the gap between PY and FY appears to be closing rapidly. Current genetic progress is linked to increased bio-mass accumulation, and this will remain the way for-ward in the future given the limits to increased harvest index (HI). There is evidence that recent progress is related to increased photosynthesis (e.g., greater radi-ation use effi ciency (RUE) at the canopy level and/or maximum photosynthetic rate P max at saturating irra-diance at the leaf level) before and around anthesis. There is no theoretical reason why this trend cannot continue, especially given the vast genetic resources already found within each crop species. However, it will not be easily or cheaply accomplished, so pros-pects for higher rates of potential yield growth appear to be limited, notwithstanding new molecular tools and claims to the contrary. Closing the yield gap, therefore, becomes more important. Many factors are involved, but breeding can also help farmers achieve this through, for example, improved host plant resistance.