We have combined Hipparcos proper-motion and parallax data for nearby stars with ground-based radial velocity measurements to find stars that may have passed (or will pass) close enough to the Sun to perturb the Oort cloud. Close stellar encounters could deflect large numbers of comets into the inner solar system, which would increase the impact hazard at Earth. We find that the rate of close approaches by star systems (single or multiple stars) within a distance D (in parsecs) from the Sun is given by N= 3.5D^2.12 Myr^-1, less than the number predicted by a simple stellar dynamics model. However, this value is clearly a lower limit because of observational incompleteness in the Hipparcos data set. One star, Gliese 710, is estimated to have a closest approach of less than 0.4 pc 1.4 Myr in the future, and several stars come within 1 pc during a +/-10 Myr interval. We have performed dynamical simulations that show that none of the passing stars perturb the Oort cloud sufficiently to create a substantial increase in the long-period comet flux at Earth's orbit.