Over the past two decades, the causal relationship between climate change and migration has gained increasing prominence on the international political agenda. Despite recent advances in both conceptual frameworks and applied techniques, the empirical evidence does not provide clear-cut conclusions, mainly due to the intrinsic complexity of the phenomena of interest, the irreducible heterogeneity of the transmission mechanisms, some common misconceptions, and, in particular, the paucity of adequate data. This data-oriented review first summarizes the findings of the most recent empirical literature and identifies the main insights as well as the most important mediating channels and contextual factors. Then, it discusses open issues and assesses the main data gaps that currently prevent more robust quantifications. Finally, the paper highlights opportunities for exploring these research questions, exploiting the potential of the existing multi-topic and multi-purpose household survey data sets, such as those produced by the World Bank's Living Standards Measurement Study. The paper focuses on the Living Standards Measurement Study-Integrated Surveys on Agriculture program to discuss potential improvements for integrating standard household surveys with additional modules and data sources.