The appeal began with Assange's defence lawyer telling a British court the case against him was legally flawed. In the February ruling in a lower court judge rejected his defence's arguments that he would face an unfair trial in Sweden.
The 40-year-old Australian's lawyer, Ben Emmerson, said the conduct described in the European arrest warrant issued by Sweden "fails to provide a fair, proper and accurate description of what is alleged against the appellant."
He said that Sweden was seeking his extradition for questioning over the claims of rape and sexual molestation by two women "not for the purpose of prosecution".
Emmerson also argued that Assange was a victim of a "philosophical and judicial mismatch" between English and Swedish law on what constituted sex crimes.
Assange arrived at the court in central London in a black car and refused to answer questions about the appeal from a scrum of cameramen and journalists. He was wearing a grey suit, blue tie, white shirt and glasses.
A supporter yelled: "Keep fighting the American empire, Julian." Other supporters including campaigning journalist John Pilger also arrived at the court.
The hearing on Tuesday and Wednesday will take place before two judges. The decision is expected to be deferred until a later date.
His Swedish lawyer, Björn Hurtig, said Tuesday he believes there are strong chances the court will decide to extradite his client.
"I think there is a big risk he will come here," Hurtig told Sveriges Radio.
"I hope this is going to go as Julian wants but I believe the chances for change are fairly small," he said, adding he thought Assange's British legal team would concentrate more this time on formalities than on criticising the Swedish legal system and Assange's accusers.
Hurtig was censured by the judge in the previous case for being "unreliable" and for making a deliberate attempt to mislead the court.
Even if the ruling goes against him, Assange's lawyers have signalled he is prepared for a lengthy legal battle and could take his challenge all the way to the Supreme Court.
The former computer hacker has been living under strict bail conditions, including wearing an electronic ankle tag and a curfew, at a friend's mansion in eastern England since December.
He was arrested in December after two Swedish women accused him of sexual assault, allegations that Assange denies.
The Australian's whistleblowing website was in the process of releasing a huge cache of leaked US diplomatic cables at the time.
It was the site's latest dump of US government documents, following the release of secret military files about the wars in Afghanistan and Iraq, and angered Washington.
Swedish authorities want to question him over the sex assault claims, although he has not been formally charged. He has claimed that the allegations are politically motivated.
Assange has shaken up his legal team ahead of the hearing, replacing vocal media lawyer Mark Stephens with Gareth Peirce, a high-profile human rights lawyer.
In recent years Peirce has represented Muslim terror suspects, and a former Guantanamo Bay inmate. She made her name in the 1970s and 80s defending people wrongfully jailed over bombings by Northern Irish paramilitary group the IRA.
Despite the tight restrictions on him, Assange managed to celebrate Sunday his 40th birthday, which occurred on July 3rd, with a party.
Assange has claimed his greatest fear is eventual extradition to the United States, where his lawyers argued he could be sent to the Guantanamo Bay detention facility or face the death penalty.
US authorities opened a criminal investigation against Assange in July 2010 but are yet to bring any charges against him.