We calculate the space motions of 21,497 stars to search for close stellar encounters with Vega, epsilon Eridani and Fomalhaut during the past 1 Myr. We discover that epsilon Eridani experienced three <2 pc encounters over the past 100,000 yr. Within the uncertainties, epsilon Eridani is having a close encounter with Kapteyn's star near the present epoch, with a 42.2 percent probability that the closest approach distance is <1 pc. Vega and Fomalhaut experienced four and six <2 pc encounters, respectively, over the past 1 Myr. Each had one encounter with a roughly 2 percent probability that the closest approach distance is less than 0.5 pc. These encounters will not directly influence the debris disks observed around Vega, epsilon Eridani and Fomalhaut, but they may pass through hypothetical Oort clouds surrounding these stars. We find that two other Vega-like stars, HD 17848 and HD 20010, experienced rare, <0.1 pc stellar encounters that are more likely to directly perturb their circumstellar disks.