After he edged out Gelfand in a very close affair in 2012, Anand didn't have too much time to rest on his laurels – he had to face a new challenger already next year in 2013. And whom? Carlsen himself! Of course, at the time, back in 2013, we didn't know yet what kind of big champion will Carlsen be, but he was always an immense talent and a big hope for the chess world.
Carlsen managed to win the famous Candidates Tournament (best 8 players) earlier that year. But it was not a smooth sail. It was very tense until the very end, and Carlsen and Kramnik went into the last round with 8 points. In the last round, they both over-pushed and both lost! Carlsen against Svidler and Kramnik against Ivanchuk. But Carlsen won on tiebreak! What a tournament. No wonder that some time ago, Carlsen said in a live broadcast that it is easier to play a match for the world championship than to win a Candidates' tournament!
The match was played from 9th till 22nd of November, 2013. In Chennai, India. Carlsen had no problem going to play on Anand's home ground. And he won!
It was planned to be 12 games match, but Carlsen won with 3-0, and 7 draws, so the last 2 games were unnecessary.
Carlsen became champion just before his 23rd birthday (30th of November), while Anand was already almost 44 years old. And how big a champion is Anand! He proved it by playing again next year in a new match – this time as the challenger. And no, it was not a rematch; he won the Candidates' Tournament in 2014. He was almost 45 years old! And he was the oldest player who played in a world championship match, since Korchnoi faced Karpov in their 2nd match for the world title, back in Merano – Italy, in 1981. Korchnoi was 50 years old.
So, this Carlsen's win of the world title was proclaimed by Kasparov as the start of a new era in chess. And indeed, after 8 years so far, we can confirm, sure it was: Carlsen is still the champion!
But Carlsen also had some tough matches. First, he had to defend his title against Anand in 2014. Which he did, with 3-1 and 7 draws.
Then he had 2 very tough matches: vs. Karjakin in 2016 and then vs. Caruana in 2018.
Against Karjakin in 2016, he won only after a tie-break because the match finished in 1-1 with 10 draws!