* Protesters, riot police converge at airport
* Police, protesters in violent overnight clashes
* Officers fired warning shots in the air on Saturday (Adds details, background, comment)
By Kai Pfaffenbach and Farah Master
HONG KONG, Sept 1 (Reuters) - Hundreds of protesters gathered outside Hong Kong's international airport on Sunday, disrupting travel in a bid to focus global attention on their fight for greater democracy in a city facing its biggest political crisis in decades.
Trains to the airport were suspended and traffic was congested after protesters urged the public to overwhelm road and rail links to one of the world's busiest airports.
But passengers continued to enter and leave the terminal freely, and planes were still taking off. Rain fell as a tropical depression approached Hong Kong from the southeast.
Authorities raised the number three typhoon signal, warning of a possible heavy storm on its way.
Police warned the airport gathering was illegal and said they would soon conduct a "dispersal operation".
"The police warn all protesters to stop their illegal acts and leave immediately," they said in a statement.
Sunday's demonstration comes after police and protesters clashed overnight in some of the most intense violence since unrest erupted more than three months ago over concerns Beijing was planning to erode the autonomy granted to the city when it was handed back to China from Britain in 1997.
China denies the charge of meddling in Hong Kong, which it says is an internal affair. It has denounced the protests and warned of the damage to the economy.
Protesters outside the airport, many wearing hard hats and gas masks, chanted "Fight for freedom! Stand with Hong Kong!" as riot police watched from inside the terminal building.
"We plan to disrupt activity at the airport to draw attention to what the government and the police are doing to us," said one 20-year-old protester, asking not to be named.
"If we disrupt the airport more foreigners will read the news about Hong Kong."
Three weeks ago, some flights were delayed or cancelled after protesters swarmed the airport.
Several hundred demonstrators also gathered outside the British consulate in central Hong Kong, waving Union Jack flags and chanting "God save the Queen".
During a chaotic night of violence into the early hours of Sunday, police fired tear gas, water cannon and rubber bullets and protesters threw petrol bombs.
As government helicopters hovered overhead, protesters who had been banned from demonstrating set fires in the streets and threw bricks at police near government offices and Chinese military headquarters.
CHALLENGE TO CHINA
Officers fired two warning shots in the air to scare off a group of protesters who had surrounded them and were trying to steal their pistols, the police said, only the second time live rounds have been used in more than three months of unrest.
Police sprayed demonstrators with blue-dyed water to make it easier to identify them later.
Parts of the metro system ground to a halt as skirmishes spread to the subway, with television showing images of people being beaten as they cowered on the floor behind umbrellas.
Police said they arrested 40 people inside Prince Edward metro station on suspicion of obstructing officers, unlawful assembly and criminal damage. Three stations stayed shut on Sunday.
"The level of violence is rapidly escalating, and their illegal acts have no regard to the laws of Hong Kong," police said in a statement that condemned unlawful assembly of protesters in various districts.
The people of Hong Kong were caught in a humanitarian crisis, protesters said in a statement on social media that appealed for international help.
It described the police crackdown at the Prince Edward station as a "terrorist attack".
"They beat up innocent people causing them serious wounds," protesters said in the statement, signed 'Hongkongers'.
"Some passengers kneeled before them asking them to stop but...were beaten even harder."
The latest protests came on the fifth anniversary of China's decision to curtail democratic reforms and rule out universal suffrage in Hong Kong, a former British colony.
The unrest began in mid-June, fuelled by anger over a now-suspended extradition bill that would have allowed people in the city to be sent to mainland China for trial in courts controlled by the Communist Party.
But the turmoil has evolved over 13 weeks to become a widespread demand for greater democracy in a territory that many residents believe is increasingly being controlled by Beijing.
Hong Kong returned to China under a "one country, two systems" formula that allows it freedoms not enjoyed on the mainland, such as the right to protest and an independent legal system.
The unrest poses the gravest challenge to Chinese President Xi Jinping since he took power in 2012, with his government keen to end the protests before the 70th anniversary of the founding of the People's Republic of China on Oct 1.
Beijing has also accused foreign powers, particularly the United States and Britain, of fomenting the demonstrations. (Reporting by Twinnie Siu, Marius Zaharia, Farah Master, Joyce Zhou, Donny Kwok, Yoyo Chow and Anne Marie Roantree; Writing by Joe Brock; Editing by Clarence Fernandez and Kenneth Maxwell)
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