Numerical simulations of dwarf spheroidal galaxies undergoing several close encounters with the Milky Way are described. By comparing our models to observed properties of the recently discovered dwarf galaxy in Sagittarius (Sgr), we discuss implications of our results for the formation and evolution of the Milky Way system. We find that existing observations are not sufficient to allow us to place precise limits on either the orbit or the initial state of the dwarf. Debris from the ongoing tidal stripping of the Sagittarius galaxy are expected to form moving groups in the halo of the Galaxy and the discovery of such stars would strongly constrain the history and dynamical state of the dwarf. Furthermore, if Sgr is presently being disrupted, we predict that its remains will be detectable as a moving group in the halo for more than 1 Gyr. Thus, if similar accretion events have occurred in the recent history of the Galaxy, their aftereffects may still be observable.

15 pages text and 6 figures in uuencoded compressed Postscript file (270 kb). Missing figures (4-7) available on request from kvj@lick.ucsc.edu