HONG KONG — What started in June with peaceful rallies in opposition to contentious legislation has devolved into a steady stream of mayhem, with some protesters embracing violent behaviours in response to brutal police tactics.

Protesters and the Hong Kong government, backed by Beijing, appear to have intractable differences, and there have been few signs of either side backing down.

In the months since the demonstrations began, the unpopular extradition bill has been withdrawn, but protesters’ demands have expanded to include increased democracy and an investigation of police.

The standoff has taken on international importance. China has viewed the protests as a challenge to its fervent nationalism, while democracy supporters worldwide have cheered what they see as a poke in the eye of the autocratic Chinese government.

It all comes amid a rancorous trade war between China and the United States, and some international businesses, including the NBA, have found themselves stuck in a political mess they wanted no part of.

Here’s a guide to how we got here and why violence on both sides has escalated.

WHAT IS HONG KONG’S RELATIONSHIP WITH CHINA?

Hong Kong, an international finance hub on China’s southern coast with more than 7 million residents, was a British colony until 1997, when it was handed back to China under a policy known as one country, two systems.

The policy made Hong Kong part of China but let it keep many liberties denied to citizens on the mainland, including free speech, unrestricted internet access and the right to free assembly. The territory has its own laws, system of government and police force under a miniconstitution known as the Basic Law. China promised that this system would remain in place until at least 2047.

But many Hong Kongers feel that Beijing is already chipping away at the city’s autonomy and that the local government does its bidding. The territory’s top leader, the chief executive — currently Mrs Carrie Lam — is appointed by a pro-Beijing committee.

WHAT DO PROTESTERS WANT?

Many people in the territory feel deep contempt for the Chinese government and hope to preserve their freedoms for as long as possible. At its core, the movement is aimed at resisting encroachment from the mainland — but it has been complicated by rising violence.

At first, the movement was focused on a bill, since scrapped, that would have allowed people accused of crimes to be sent to places with which Hong Kong had no extradition treaty — including mainland China, where courts are controlled by the Communist Party.

Hundreds of thousands of people, fearing the bill would allow Beijing to target dissidents in Hong Kong with phony charges, joined a peaceful march to oppose the bill June 9.

On June 12, for the first time, police used pepper spray, batons and more than 150 canisters of tear gas to disperse thousands of protesters, a small number of whom had thrown projectiles at police. Irate at the police response, protesters demanded an independent investigation of the police force — a demand leaders have refused.

While anger at police has been a driving force, protesters have extended their demands to include amnesty for arrested participants and direct elections for all lawmakers and the chief executive. The movement is largely leaderless, with decisions frequently made through voting in online forums.

WHY HAVE THE DEMONSTRATIONS TURNED VIOLENT?

A vast majority of participants have been nonviolent, staging strikes, surrounding police stations, shutting down the airport and forming huge marches. The city’s creative class has turned protest into art and song.

But a minority of protesters has become increasingly destructive, hoping to force the government’s hand. Since only one of the protesters’ demands has been met — the withdrawal of the extradition bill — the more violent participants felt peaceful rallies were ineffective.

Some protesters have thrown bricks and Molotov cocktails and in one case stabbed a police officer. Police said that one homemade bomb was detonated during a protest. On several occasions, protesters have doled out vigilante justice, beating people who were perceived to be against their movement, including one man who was doused with fluid and set on fire.

A firebrand pro-Beijing lawmaker, Mr Junius Ho, was attacked with a knife. There has been considerable property damage to the train system, which protesters have accused of supporting police, and businesses seen as pro-China.

Protesters said the violence is in defense of excessively violent police tactics. Officers have liberally deployed tear gas in ways that defy international standards, including firing canisters from a height and using it in enclosed spaces. Several protesters have been shot with live rounds.

Cannons shooting water laced with a corrosive blue dye have become a routine presence at protests. Individual interactions with police that are captured on video, such as the use of pepper spray against a pregnant woman and an officer on a motorcycle swerving into protesters, have infuriated demonstrators.

Passions were further inflamed when a Hong Kong student died after falling from a parking garage near demonstrations, possibly the first death of the movement. The exact circumstances surrounding his death remain unknown.

Recent standoffs on university campuses have resembled trench warfare, with officers firing tear gas and protesters volleying with firebombs and other projectiles.

HOW DOES IT END?

No one knows.

But despite some domestic propaganda showing tanks assembling across the border in Beijing, it appears China is trying to avoid a Tiananmen-style crackdown. Although the Chinese military has a garrison in Hong Kong, the international business community would most likely see a military intervention as the end of “one country, two systems,” and an exodus of businesses could soon follow.

Still, China does not want to bend to protesters, whom the state news media have depicted as lawless, spoiled separatists. (Most protesters said they are uninterested in independence.)

The Hong Kong government and police have done little to calm tensions, repeatedly denouncing protesters while mostly defending police conduct. And protesters have shown few signs of fatigue, despite thousands of arrests. THE NEW YORK TIMES