Last month, a group of 10 families filed an action against the European Union at the European General Court, the EU’s second highest court. All claim to have suffered loss from climate change. Specifically, they argue that greenhouse gas (GHG) emissions from the EU have violated their fundamental rights to health, occupation, property and equal treatment.

The plaintiffs want the EU to tighten its greenhouse gas reduction targets by building on the successful 2015 Urgenda case, where the Dutch courts obliged the government to act more forcefully to fight climate change. The Dutch government has appealed, but the case has already inspired copycat suits the world over.

Climate change lawsuits are on the rise, and not just against governments. A big target is the oil majors. Cities such as New York and San Francisco in the U.S. allege that these companies continued to produce and market fossil fuels while knowingly concealing climate risks. They are also being sued by private plaintiffs for being a public nuisance, for failure to warn about climate risks, climate-related negligence and trespassing.

Even if most of the more than 1,000 climate lawsuits outstanding in 28 national and seven international jurisdictions fail, they contribute to raising awareness that neither governments nor oil companies are doing enough to address climate change. They also make investors and regulators more aware of the risks to fossil fuel firms from both stricter policies and the potential award of damages.

The process of “discovery,” already underway in a number of the suits, is likely to bring new and potentially damaging information to light, further tilting the scales of public opinion, risk perception and future legal liability against the fossil fuel majors.

And some may not fail. Risks from climate lawsuits for fossil fuel firms are likely to increase sharply—and they may soon start crystallizing into costs, for a number of reasons.

First, the number of national and international statutes pertaining to climate change continues to rise, with 106 new laws already having been passed since the Paris agreement, including 28 that explicitly reference it. Adding in existing laws and executive polices takes that to 2,000. The more laws there are, the more lawsuits and legal routes to remedy to expect.

Second, each successful lawsuit creates a precedent, which others can often build on. Even when the precedent is in other jurisdictions, it can help inspire plaintiffs bring similar suits. Successful climate change lawsuits will also help change jurisprudence, even where precedent may not apply directly.

Third, the damage from climate change is accelerating, with the frequency of extreme weather events having risen five-fold in the past 30 to 40 years, for example. Rating agencies such as Moody’s are already starting to factor exposure to climate change in their ratings. Expect an increasing number of litigants to seek compensation.

Fourth, the science of attributing specific damage, for example, from a flood, to climate change, has improved exponentially in the past decade. Scientists are increasingly ready to serve as expert witnesses. Courts already regularly accept probabilistic arguments, such as on exposure to hazardous chemicals. Causality linking damage and harm to climate change can now be established with increasing confidence and robustness.

Fifth, advances in measurements such as the development of the Carbon Majors database means that the proportional culpability of specific listed firms, such as Exxon Mobil, can be reliably established through their share of cumulative emissions.

Every small victory sets an important precedent, which future lawsuits can build on.

Many of the earliest lawsuits against Big Tobacco also failed, but they were critical in helping to establish precedents on scientific consensus, on the harm being caused and on the duplicity of tobacco companies that had actively suppressed research thus misleading the public on the harmful effects of smoking.

While successes have been few thus far, the attitudes of judges shift with the times. An international panel of senior judges admitted that judges are influenced by the zeitgeist of growing worries and mounting evidence about climate change, so are more likely to look favorably at climate liability lawsuits. It concluded that, in fact, many companies may already be in violation of existing laws on climate change.

Some oil majors are finally starting to acknowledge the rising risk. In a recent filing with the Securities and Exchange Commission Chevron admitted that climate litigation risks could have a “material adverse effect on the company,” “curtail profitability” and even render the business model of carbon majors such as itself “economically infeasible.” In a rather bizarre exchange with shareholders, BP’s CEO Bob Dudley refused to disclose certain climate targets, or even answer some questions from activist investors, citing a very real threat from class-action lawsuits in the U.S.

Last week, Legal & General, Europe’s second largest asset manager, pledged to vote against companies that were falling behind on action to tackle climate change. The boards of fossil fuel majors and the asset managers that are major shareholders have a duty to be more transparent about the risks they are facing and offer strategies for managing them. So far, they have shown mainly reluctance.

( To contact the author of this story: Sony Kapoor at Bloomberg View Column: This column does not necessarily reflect the opinion of the editorial board or Bloomberg LP and its owners.)