Abstract We develop a general theory of omitted variable bias for a wide range of common causal parameters, including average treatment effects, average causal derivatives, and policy effects from covariate shifts. We show how plausibility judgments on the maximum explanatory power of omitted variables are sufficient to bound the bias, facilitating sensitivity analysis in otherwise complex models. Finally, we provide statistical inference methods that can leverage modern machine learning algorithms for estimation. These results allow empirical researchers to perform sensitivity analyses in a flexible class of machine-learned causal models using very simple tools. Empirical examples demonstrate the utility of our approach.