After Fischer didn't show up for the match title in 1975, in Manilla, Karpov became the world champion by forfeit.
Anatoly Karpov in 1978
Anatoly then played a lot of tournaments and had a lot of success, proving that he was indeed the world's best player.
Then he faced Korchnoi in 2 matches for the world title. In Baguio in 1978 and in Merano in 1981. It is worth mentioning that Korchnoi is alongside Keres and Bronshtein, one of the best players in history who never became world champion.
The match in Baguio in 1978 was played from July 18th till October 18th. It was a "real deal" because Karpov was first in the FIDE rating list, with Elo 2700, and Korchnoi was second, with Elo 2695.
The match was tough and controversial. Some sources even say that those 2 matches between Karpov and Korchnoi were "matches of hatred." Also interesting is that Karpov was 27 years old, and Korchnoi 47. But in those years, Korchnoi was the world's second player, as he also beat Spassky in the Candidates final match (which determined the challenger to Karpov). And again, in the next cycle, Korchnoi became challenger, this time being 50 years old!
Karpov was born in 1951 and became GM at 19 years of age. Later on, his Elo peak was at 2780! He won around 160 tournaments and was in the top spot of the rating FIDE list for 102 months (the list exists since 1970.)
Korchnoi was no joke either. He was born in 1931 in Leningrad. Grandmaster since 1956. Then he fled to the Netherlands in 1976 and finally settled in Switzerland in 1978. He died in 2016.
I was lucky enough to play 2 classical games against him; the first in San Sebastian in 2011 and the second in Bad Woerishofen in 2012. Both finished in a draw. Korchnoi spoke several languages fluently, and he spoke with me after both games in perfect Serbian language!
In Baguio, Karpov had the lead with 4-1 and then 5-2, but Korchnoi struck back, and he leveled the score to 5-5, after 31 games (21 draws). They played till 6 wins (player who first wins 6 games, wins the match), and Karpov won the very next game, thus winning the match with 6-5, with 21 draws. Tough one.
The next match they played, in Merano in 1981, it was easier for Karpov, and that time he won 6-2 with 10 draws.
And then came the famous 5 Kasparov-Karpov matches, 1984-1990.