Who is the world's most popular chess player?

The world's most popular chess player in the world was Bobby Fischer. Nowadays Garry Kasparov is very popular as well.

EDIT: All true. However, Kasparov no longer plays competitive chess so his popularity, as far as a chess player, has waned, and Fischer is, well, dead.

Currently the most popular Grandmasters are Vaselin Topalov, Vishy Anand, and Magnus Carlsen, and, if you are from the USA, Hikaru Nakamura.

Currently, Vishy Anand is the reigning World Chess Champion recognized by FIDE. He won the undisputed title in 2008 against Vladimir Kramnik. This match was necessary as Anand was crowned the champion via being the victor in a Championship Tournament, not as a result of Match Play. The result against Kramnik put that 'dispute' to rest.

Vaselin Topalov defeated Gata Kamsky to win the right to challenge Anand in the next championship match, which has yet to be scheduled in its finality.

also Kasporov