The World Champions of Chess

Before the official World Champions  
Giaochino Greco 1620-34 Italy

Chess existed a long while before Greco came along, but he was pretty clearly the best chess player of his time and worth knowing about. 

There were all sorts of players after that who might have been the best in the world, like Philidor, Saint-Amant, and Staunton.

We pick up the story a long while later, with some players who equally clearly were the best of their time:  

Adolf Anderssen 1851–58,1862–66 Prussia
Paul Morphy 1858–62 United States
World Champions  
In 1886, Wilhelm Steinitz beat Johannes Zukertort in a match which was agreed to be for the title of World Chess Champion.  Steinitz had previously beaten Anderssen in a match in 1866, so he may have been the best in the world since then.  
Wilhelm Steinitz 1886–94 Austria-Hungary, USA
(Hastings 1895 a strong tournament)
Emanuel Lasker 1894–1921 Germany
(Carl Schlechter of Austria-Hungary drew a match with Lasker) 1910
José Raúl Capablanca 1921–27 Cuba
Alexander Alekhine 1927–35,1937-1948
Max Euwe 1935–37 Netherlands
Mikhail Botvinnik 1948–57, 58–60, 61–63 Soviet Union (Russia)
(David Bronstein of the USSR drew a match with Botvinnik) 1951 
Vasily Smyslov 1957–58 Soviet Union (Russia) (Drew match in 1954)
(Botvinnik again) 1958-60
Mikhail Tal 1960–61 Soviet Union (Latvia)
(Botvinnik again) 1961-63
Tigran Petrosian 1963–69 Soviet Union (Armenia)
Boris Spassky 1969–72 Soviet Union (Russia)
Robert Fischer 1972–75 United States
Anatoly Karpov 1975–85 Soviet Union (Russia) (and drew in 1987)
Garry Kasparov 1985–93*,1993–2000 Soviet Union (Russia)
Vladimir Kramnik 2000–07 Russia
(Peter Leko of Hungary drew a match) 2004
Viswanathan Anand 2007–13 India
Magnus Carlsen 2013–present Norway
Not World Champions  
Xavery Tartakower  Poland, Austria, France  (The world champion of chess writing)
Judit Polgar  Hungary (The best female player ever)

In 1993*, the World Champion, Garry Kasparov, decided to organise his own match, and so did FIDE, and so there were two World Champions from 1993-2006, but Kasparov was definitely the best player until 2000, and Kramnik, who beat him, later beat Topalov, who was then the FIDE Champion, in a unification match in 2006.