A country's performance in health attainment refers to both its achievement (level) and its improvement (evolution) in the health domain. Studies on performance generally measure health attainment using the average health level of the population, and quantify health improvement employing the change in attainment over time. However this approach is flawed because the change in attainment does not satisfy good properties, on the one hand, and because health attainment should not only account for the average health level, but also for disparities in health in the population, on the other hand. We propose a solution to the first limitation by following the lead of Kakwani (1993), who uses achievement and improvement measures which are based on attainment measures and which satisfy important properties. For the second limitation, we extend the work of Kakwani and propose new definitions of attainment that account for the average health level but also for health inequalities in the population. Specifically, we focus on overall and social health inequalities and on the health of the poor. By including these new attainment variables into Kakwani's indices, we generate new classes of achievement and improvement indices. Using data on 11 low and middle-income Asian countries in the twenty-first century, we highlight that child and maternal health have generally improved in recent decades, due to both an increase in the average health level and a decrease in inequalities.