Exxon Mobil dodged a bullet last month when a judge rejected a novel climate-change lawsuit brought by New York’s attorney general. The case began with a promise from state officials that there would be a historic reckoning for the fossil fuel giant.  It ended ignominiously as a failed accounting fraud claim. But that was just the beginning. Globally, humans are on the hook for trillions of dollars if they want to sufficiently reduce greenhouse gas emissions, acclimate to the damage already done and prepare for what is yet to come. As more governments and taxpayers find themselves staring down the barrel at rising climate costs, they are increasingly turning to the courts to hold Big Oil accountable.

The New York case was an outlier—it sought to make Exxon Mobil investors whole for an alleged bookkeeping bait-and-switch. The majority of U.S. climate litigation out there takes a more direct approach, seeking damages in so-called public nuisance lawsuits. Fossil fuel use runs counter to the inherent right to exist in a non-warming world, the argument goes, and the energy companies knew that right would be infringed when enough of it was burned.

About a dozen cities, counties and states have sued Exxon, Chevron, BP, Royal Dutch Shell and their peers. The suits seek to reimburse taxpayers for the costs associated with adapting to climate change—from building multibillion-dollar sea walls to repairing damage from powerful storms and, perhaps soon, moving whole communities inland. Federal appeals courts on both sides of the country are considering whether such cases may proceed. Their rulings—one of which may come any day—will have a powerful effect on the future of climate change litigation.