The "shooting stars" will be seen primarily from the Northern Hemisphere, with the best views predicted from the early hours of Tuesday, 19 November.
How to see the meteor shower
Choose a dark location, away from city lights
Look up towards an unobstructed part of the sky
Face away from the Moon
If enthusiasts are lucky enough to experience clear skies, then they could see as many as two or three bright meteors a minute.
The shooting stars are actually particles from Comet Tempel-Tuttle which have a 33-year orbit around the Sun.
As the Earth passes through the cloud of dust left by the comet, these particles burn up in the atmosphere, creating the light streaks.
Look east
Astronomers predict that this is likely to be the most intense shower for many years to come.
Despite the fullness of the Moon, this year's Leonid shower should be very visible provided cloud does not obscure the show.
In Northern Europe, the area to watch is the stretch of sky between the north-east and south-east between 0300 and 0500 GMT.
While in North America the burst is expected to start at 1030 GMT (0530 EST or 0230 PST).
Bille Cooke, a meteor forecaster from the US space agency's (Nasa) Marshall Space Flight Center, said: "Try and get away from city lights; the darker the sky, the more meteors you'll see."
And there is no guarantee that the shower will appear on cue.
Little bullets
Robin Scagell, the vice-president of the Society for Popular Astronomy in the UK, said: "If we do get a meteor storm, it should be worth setting the alarm for.
"But you will need to watch for some time to stand a good chance of seeing the meteors."
Professor Mark Bailey of Armagh Observatory, in Northern Ireland, recommends looking out for the meteors from 2330 GMT on Monday, 18 November, onwards.
"There should be increasing numbers after 0200, reaching a peak about two hours later, then dropping once more," he said.
"However, there is always the chance of a surprise outburst, as we had a few years ago."
Satellite warning
Despite the tiny size of the Leonid particles, their speed relative to the Earth makes them a hazard to any equipment orbiting outside the protection of the atmosphere.
The European Space Agency has issued a warning to satellite operators to move their equipment to present the narrowest possible profile to the direction of the shower, and to power down any sensitive equipment.
The agency has also recommended delaying spacecraft launches planned for this period.