Objective To advance the goal of "Grand Convergence" in global health by 2035, this study tested the convergence hypothesis in the progress of the health status of individuals from 193 countries, using both standard and cutting-edge convergence metrics. Methods The study used multiple data sources. The methods section is categorized into two parts. (1) Health inequality measures were used for estimating inter-country inequalities. Dispersion Measure of Mortality (DMM) is used for measuring absolute inequality and Gini Coefficient for relative inequality. (2) We tested the standard convergence hypothesis for the progress in Infant Mortality Rate (IMR) and Life Expectancy at Birth (LEB) during 1950 to 2015 using methods ranging from simple graphical tools (catching-up plots) to standard parametric (absolute [beta] and [sigma]-convergence) and nonparametric econometric models (kernel density estimates) to detect the presence of convergence (or divergence) and convergence clubs. Findings The findings lend support to the "rise and fall" of world health inequalities measured using Life Expectancy at Birth (LEB) and Infant Mortality Rate (IMR). The test of absolute [beta]-convergence for the entire period and in the recent period supports the convergence hypothesis for LEB ([beta] = -0.0210 [95% CI -0.0227 - -0.0194], p Conclusion We found that with a current rate of progress (2.2% per annum) the "Grand convergence" in global health can be achieved only by 2060 instead of 2035. We suggest that a roadmap to achieve "Grand Convergence" in global health should include more radical changes and work for increasing efficiency with equity to achieve a "Grand convergence" in health status across the countries by 2035.