This paper provides a framework for understanding the cross-section and time series approaches used to test the convergence hypothesis. It defines two notions of convergence: one based on the tendency of output differences to narrow over a fixed time period (catching up), and another based on the long-run equality of output differences (equality of long-term forecasts). These definitions are implications of the neoclassical growth model, which predicts that economies will converge to the same level of per capita output regardless of initial conditions. However, new growth theories challenge this, suggesting that nonconvexities in production can lead to multiple equilibria and prevent convergence. Cross-section tests examine the correlation between initial per capita output and subsequent growth rates, while time series tests focus on the long-run behavior of output differences. Cross-section tests generally reject the no-convergence null hypothesis for advanced economies and US regions, whereas time series tests often accept it. The paper shows that cross-section tests are based on a weaker notion of convergence than time series tests and may fail to distinguish between neoclassical and new growth models. Time series tests, on the other hand, assume that data are well-characterized by a limiting distribution and may be invalid if data are far from this distribution. The paper also highlights the different assumptions underlying cross-section and time series tests. Cross-section tests assume data are in transition towards a limiting distribution, while time series tests assume data are near their limiting distributions. This leads to different interpretations of convergence. Cross-section tests may reject the no-convergence null for data from economies with different long-run steady states, while time series tests may accept it for data from multiple equilibria. However, time series tests may be invalid if data are driven by transition dynamics. The paper concludes that neither testing framework is likely to yield unambiguous conclusions about competing growth models. Cross-section evidence can be consistent with new growth theory, while time series results accepting the no-convergence null may reflect transitional dynamics. A more general empirical methodology would integrate transition and steady-state information to better analyze convergence.This paper provides a framework for understanding the cross-section and time series approaches used to test the convergence hypothesis. It defines two notions of convergence: one based on the tendency of output differences to narrow over a fixed time period (catching up), and another based on the long-run equality of output differences (equality of long-term forecasts). These definitions are implications of the neoclassical growth model, which predicts that economies will converge to the same level of per capita output regardless of initial conditions. However, new growth theories challenge this, suggesting that nonconvexities in production can lead to multiple equilibria and prevent convergence. Cross-section tests examine the correlation between initial per capita output and subsequent growth rates, while time series tests focus on the long-run behavior of output differences. Cross-section tests generally reject the no-convergence null hypothesis for advanced economies and US regions, whereas time series tests often accept it. The paper shows that cross-section tests are based on a weaker notion of convergence than time series tests and may fail to distinguish between neoclassical and new growth models. Time series tests, on the other hand, assume that data are well-characterized by a limiting distribution and may be invalid if data are far from this distribution. The paper also highlights the different assumptions underlying cross-section and time series tests. Cross-section tests assume data are in transition towards a limiting distribution, while time series tests assume data are near their limiting distributions. This leads to different interpretations of convergence. Cross-section tests may reject the no-convergence null for data from economies with different long-run steady states, while time series tests may accept it for data from multiple equilibria. However, time series tests may be invalid if data are driven by transition dynamics. The paper concludes that neither testing framework is likely to yield unambiguous conclusions about competing growth models. Cross-section evidence can be consistent with new growth theory, while time series results accepting the no-convergence null may reflect transitional dynamics. A more general empirical methodology would integrate transition and steady-state information to better analyze convergence.