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We present and solve a continuum theory of vacancy-mediated diffusion (as evidenced, for example, in the vacancy driven motion of tracers in crystals). We restrict our attention to ultralow vacancy densities, so that a tracer only experiences a single vacancy over the time scales of interest. Results are obtained for all spatial dimensions, and reveal the strongly non-Gaussian nature of the tracer fluctuations. In integer dimensions, our results are in complete agreement with those from previous exact lattice calculations. We also extend our model to describe the vacancy-driven fluctuations of a slaved flux line. [S0163-1829(99)02331-0].