New World

The New World is the ideal society Light Yagami endeavors to create through the use of the Death Note. He boasts about creating this world as early as the end of chapter 1 of the manga and the first episode of the anime.

Kira's New World
His plan to establish the new world is comprised of various steps. He begins by eradicating the most dangerous criminals of the world through heart attacks and then kills off petty criminals through accidents and diseases, though even some lesser criminals are also killed by heart attack. It is only after L begins to pursue him does Light begin killing innocents, namely law enforcement officials tasked with capturing him.

Light, however, would rather have the world's police force to work alongside him rather than oppose him. That is yet another aspect of his ideal world. This part of his plan does, in some ways, come into fruition when he eventually kills L and takes his place. By taking the position of L, Light not only has access to more criminal records, but has the assistance of police agencies around the world, allowing him to even have tabs on the president.

Late in the series, when Teru Mikami is serving as Light's proxy, the latter becomes baffled when he discovers that the former has deemed it necessary to kill those who refuse to contribute to the good of society. Light does remark that it is much too soon to be making such an order, implying that as soon as the “new world” was created, Light himself would advocate for the deaths of people who refused to improve and help other people.

According to chapter 100 of the manga, the new world was to begin as soon as Light had laid waste to the last of his opposition. Following this, Kira would absorb the education curriculum of the world and people would be taught to act purely on altruism and kindness. Ideally, this would eventually create a utopia.

Although Light, by the end of the series, had managed to sway most of the world’s population over to his side, he was met with opposition by the likes of Near and other anti-Kira organizations, and they eventually brought his demise.

Although the morality of Light's actions were questionable, they did arguably bear some results. Near himself even admitted in the one-shot manga special that the world under Light's short-lived regime did see the end of wars and crime rate drop 70%.

The New World in L: Change the WorLd
In the movie and novel L: Change the WorLd, the people of the Blue Ship organization, most notably K, plan on creating a new, ideal society, which they also refer to as the "New World." Rather than wanting to get rid of criminals, they believed that the problem with the current world is overpopulation, and in order to create a new, better world, the majority of mankind needed to be killed. To do this, an extremely deadly and contagious virus was created. The virus was to be released so that it would spread across the entire world, affecting everyone and killing of the majority of the planet. All that was needed was the antidote, in order to cure people who were worthy of living in the New World.

K stated that people like Professor Nikaido would be needed in the New World.

Trivia

 * Taro Kagami, the protagonist of the pilot chapter, remarks going through a similar crusade as Light did. Whether or not he actually does is open to interpretation, since, by the time the pilot has ended, he still has ownership of his notebook.