I see so much on the site about Bayesian probability. Much of my current work uses Dempster-Shafer theory, which I haven't seen mentioned here.

DST is a generalization of Bayesian probability, and both fuzzy logic and Bayesian inference can be perfectly derived from DST. The most obvious difference is that DST parameterizes confidence, so that a 0.5 prior with no support is treated differently than a 0.5 prior with good support. For my work, the more important aspect is that DST is more forgiving when my sensors lie to me; it handles conflicting evidence more gracefully, as long as its results are correctly interpreted (in my opinion they are less intuitive than strict probabilities).