Directed acyclic graphs (DAGs) are now a popular tool to inform causal inferences. We discuss how DAGs can also be used to encode theoretical assumptions about nonprobability samples and survey nonresponse and to determine whether population quantities including conditional distributions and regressions can be identified. We describe sources of bias and assumptions for eliminating it in various selection scenarios. We then introduce and analyze graphical representations of multiple selection stages in the data collection process, and highlight the strong assumptions implicit in using only design weights. Furthermore, we show that the common practice of selecting adjustment variables based on correlations with sample selection and outcome variables of interest is ill-justified and that nonresponse weighting when the interest is in causal inference may come at severe costs. Finally, we identify further areas for survey methodology research that can benefit from advances in causal graph theory.