After Tempel-Tuttle's orbit had been calculated from the 1865 apparition, astronomers quickly showed that it was the parent body of the Leonid meteor shower.
This shower, normally exhibits around 10-20 meteors per hour over a period of six or seven days, with the peak occurring on November 17. However, every 33 years, more dramatic displays tend to be observed. (In 1966, the central USA saw about 5000 meteors in a 20 minute period.) These dramatic displays tend occur in years when T-T comes to perihelion, which means that the coming years are candidates for good shows.
Be cautioned, however, that the 1998 orbital conditions of Temple-Tuttle are not as good as they were in 1965-66, so the number of meteors should not be expected to be as high as they were during the 1966 event.
Summary of the 1998 Leonid Shower
Predictions for 1999 and beyond
Observing the Leonid shower and more information