Economists are often interested in the mechanisms by which a particular treatment affects an outcome. This paper develops tests for the "sharp null of full mediation" that the treatment $D$ operates on the outcome $Y$ only through a particular conjectured mechanism (or sets of mechanisms) $M$. A key observation is that if $D$ is randomly assigned and has a monotone effect on $M$, then $D$ is a valid instrumental variable for the LATE of $M$ on $Y$. Existing tools for testing the validity of the LATE assumptions can thus be used to test the sharp null of full mediation when $M$ and $D$ are binary. We extend these results to settings where $M$ is multi-valued or multi-dimensional. We further provide methods for lower-bounding the size of the alternative mechanisms when the sharp null is rejected. 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. (joint with Jonathan Roth)