It is by now well established in the public health literature that health and nutrition in the first years of life are critical to health and wellbeing later in life. In this paper, we examine the patterns of inequality of opportunity in health and nutrition outcomes, such as height-for-age and weight-for-height, for children under 5 years of age in selected Arab Countries and Turkey, using Demographic and Health Survey (DHS) data. Our objective is to decompose inequality into a portion that is due to inequality of opportunity and a portion that is due to other factors, such as random variations in health and genetics. Inequality of opportunity is defined as the inequality that is due to differences in circumstances, such as parental characteristics, household wealth, place of birth and gender. We measure inequality using decomposable general entropy measures, such as the Theil-T index, and use parametric decomposition methods to determine the share of inequality of opportunity in total inequality. Results show that different levels and trends are evident across countries in both the overall inequality of child health outcomes and in the share of inequality of opportunity in total inequality. Inequality of opportunity is shown to contribute substantially to the inequality of child health outcomes, but its share in total inequality varies significantly both across countries and within countries over time. To further highlight the relative contribution of circumstances to the inequality of child health outcomes in different countries, we simulate height and weight outcomes for a most and least advantaged child in each context. Since these simulations set observed circumstances at their best and worst levels, the larger the difference in predicted outcomes between the most and least advantaged child, the larger is the inequality of opportunity facing children in that country.