Economists are often interested in the mechanisms by which a treatment affects an outcome. We develop tests for the "sharp null of full mediation" that a treatment $D$ affects an outcome $Y$ only through a particular mechanism (or set of mechanisms) $M$. Our approach exploits connections between mediation analysis and the econometric literature on testing instrument validity. We also provide tools for quantifying the magnitude of alternative mechanisms when the sharp null is rejected: we derive sharp lower bounds on the fraction of individuals whose outcome is affected by the treatment despite having the same value of $M$ under both treatments (``always-takers''), as well as sharp bounds on the average effect of the treatment for such always-takers. An advantage of our approach relative to existing tools for mediation analysis is that it does not require stringent assumptions about how $M$ is assigned. We illustrate our methodology in two empirical applications.

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