HOW WOULD A VEGAN WORLD BE?
According to the Poore and Nemecek study done by Oxford University and the magazine Science, we only need 25% of all land used for animal agriculture at the moment to feed everyone on earth. Animal agriculture already occupies 45% of all habitable land, so the 75% left would be a very large piece of land.
That would end global hunger, by feeding everyone with the 25% of land, and the land left could be used to create more cities, solving the overpopulation issue as well.
The number of deaths from heart disease, the current main human killer, would decrease immensely as animal products are the current leading cause of that disease. The number of deaths from diabetes and various cancers and stroke would also decrease, as animal products greatly increase the risk for those diseases. 
Deforestation would stop practically completely, as the need for more land for either food or habitation is no longer existent. 
The number of ocean dead zones, water pollution, and the overwhelming rate of species extinction would also greatly decrease, saving many species from going extinct, as animal agriculture is the main cause of those too. 
Global warming would be greatly reduced, preventing many catastrophic effects from taking place, as animal agriculture was one of the leading drivers of that issue. And, with no more deforestation, there will be more trees to help fight what is left of global warming. 
Trillions of animals murdered for unnecessary food every year will not be murdered anymore, ending that abundance of animal cruelty.
But, those animals will not overpopulate: The world will go vegan gradually, and so as less people buy these products, less animals are bred into existence to create these products, until no more animals are bred, and the few left are eaten by the few omnivores left, and the the world goes vegan, with no more farm animals except for some living in sanctuaries. It may sound sad, but it is much better than a short life full of only fear, pain, and torture.
: See sections "Animal Products and Disease" and "Veganism and Disease" on the "The Facts" page for sources.