Food security and land required for food production largely depend on rate of yield gain of major cereal crops. Previous projections of food security are often more optimistic than what historical yield trends would support. Many econometric projections of future food production assume compound rates of yield gain, which are not consistent with historical yield trends. Here we provide a framework to characterize past yield trends and show that linear trajectories adequately describe past yield trends, which means the relative rate of gain decreases over time. Furthermore, there is evidence of yield plateaus or abrupt decreases in rate of yield gain, including rice in eastern Asia and wheat in northwest Europe, which account for 31% of total global rice, wheat and maize production. Estimating future food production capacity would benefit from an analysis of past crop yield trends based on a robust statistical analysis framework that evaluates historical yield trajectories and plateaus.