We examine the hypothesis that dynamic deformations associated with seismic waves trigger earthquakes in many tectonic environments. Our analysis focuses on seismicity at close range (within the aftershock zone), complementing published studies of long-range triggering. Our results suggest that dynamic triggering is not confined to remote distances or to geothermal and volcanic regions. Long unilaterally propagating ruptures may focus radiated dynamic deformations in the propagation direction. Therefore, we expect seismicity triggered dynamically by a directive rupture to occur asymmetrically, with a majority of triggered earthquakes in the direction of rupture propagation. Bilaterally propagating ruptures also may be directive, and we propose simple criteria for assessing their directivity. We compare the inferred rupture direction and observed seismicity rate change following 15 earthquakes (M 5.7 to M 8.1) that occured in California and Idaho in the United States, the Gulf of Aqaba, Syria, Guatemala, China, New Guinea, Turkey, Japan, Mexico, and Antarctica. Nine of these mainshocks had clearly directive, unilateral ruptures. Of these nine, seven apparently induced an asymmetric increase in seismicity rate that correlates with the rupture direction. The two exceptions include an earthquake preceded by a comparable-magnitude event on a conjugate fault and another for which data limitations prohibited conclusive results. Similar (but weaker) correlations were found for the bilaterally rupturing earthquakes we studied. Although the static stress change also may trigger seismicity, it and the seismicity it triggers are expected to be similarly asymmetric only if the final slip is skewed toward the rupture terminus. For several of the directive earthquakes, we suggest that the seismicity rate change correlates better with the dynamic stress field than the static stress change.