Lawsuits against fossil fuel companies over climate change are piling up. Legislators and activists are pushing prosecutors to pursue criminal charges. Children are suing governments, arguing that their right to a healthy environment is being trampled on.

Welcome to the new universe of climate litigation, where the courts have become one of the most important battlegrounds in the fight over the greenhouse gas emissions warming the planet.

Around the world, both innovative and old-school legal arguments are being used to go after companies and governments to seek redress or forestall future harms. At the same time, the fossil fuel industry and its allies have powerful new legal grounds at their disposal to challenge climate rules. A number of cases could be taken up by the highest courts, including the U.S. Supreme Court and The Hague.

There are myriad lawsuits, which fit into several important categories. Here’s what to watch in the coming months.

Lawsuits against oil giants are spreading.

At least 86 lawsuits have been filed globally against the world’s biggest oil, gas and coal producers, according to a report published Thursday by the advocacy and research groups Oil Change International and Zero Carbon Analytics. The vast majority of those were filed in just the past nine years since the signing of the 2015 Paris accord, the landmark agreement among nations to reduce greenhouse gas emissions.