This paper proposes and tests new definitions of convergence and common trends for per capita output. We define convergence for a group of countries to mean that each country has identical long-run trends, either stochastic or deterministic, while common trends allow for proportionality of the stochastic elements. These definitions lead naturally to the use of cointegration techniques in testing. Using century-long time series for 15 OECD economies, we reject convergence but find substantial evidence for common trends. Smaller samples of European countries also reject convergence but are driven by a lower number of common stochastic trends.