We develop an omitted variable bias framework for sensitivity analysis of instrumental variable (IV) estimates that is immune to "weak instruments," naturally handles multiple "side-effects" and "confounders," exploits expert knowledge to bound sensitivity parameters, and can be easily implemented with standard software. In particular, we introduce sensitivity statistics for routine reporting, such as robustness values for IV estimates, describing the minimum strength that omitted variables need to have to change the conclusions of a study. We show how these depend upon the sensitivity of two familiar auxiliary estimates–the effect of the instrument on the treatment (the "first-stage") and the effect of the instrument on the outcome (the "reduced form")–and how an extensive set of sensitivity questions can be answered from those alone. Next, we provide tools that fully characterize the sensitivity of point-estimates and confidence intervals to violations of the standard IV assumptions. Finally, we offer formal bounds on the worst damage caused by these violations by means of comparisons with the explanatory power of observed variables. We illustrate our tools with several examples.