The Kurdish Freedom Hawks (TAK), a breakaway group of the Kurdistan Worker's Party (PKK), said in a statement on its website that the Turkish people were not the target of Saturday night's attack.
Thirteen people have have been arrested in connection with the blasts, Turkish Interior Minister Süleyman Soylu said at a press conference Sunday. Of those killed, 30 were police officers, Soylu said.
The explosions, one large blast followed by a smaller one, occurred about 11 p.m. local time (3 p.m. ET) after a heavily attended soccer game at Besiktas Vodafone Arena.
A remote control detonated a car bomb for the first explosion, according to Soylu. Shortly afterward, a suicide bomber caused a second explosion at Macka Park. The two locations are less than a mile apart.
Turkish President Recep Tayyip Erdogan, an ally to the United States in the fight against ISIS, spoke at a news conference after visiting a hospital to meet with those injured during the attacks.
"The most important topic is on how we will stand against terrorist attacks, and I want the people to know that we will fight this until it ends," Erdogan said.
"We will not let them discourage us and make us afraid of them. This country is its people. If you don't have respect and care for its people then we can not let this get away."
The Turkish prime minister's office said in a statement, a day of national mourning has been declared on Sunday, the state-run Anadolu news agency said.
Numan Kurtulmuş, Turkish Deputy PM, said the attack was planned in detail.
"The blast occurred after fans from both teams left the stadium," Kurtulmuş said. "The blast hit riot police units who were on routine duty after the game in the area. So the attack directly targeted our police."
Experts estimate at least 300-400 kilos of explosives were used in the car bomb attack, he said.
"The attack was so big that the vehicle used in bombing was not physically visible in the aftermath. It was almost completely destroyed," he said.
Video showed a chaotic scene outside Besiktas Vodafone Arena as police converged on the area and emergency medical workers loaded victims into ambulances. Several blocks away, police towed cars parked at Taksim Square, a popular tourist area, as a precaution.
Istanbul's team Besiktas had played Bursaspor in a Turkey League game that night at the football arena. The arena, which seats more than 40,000 people, had been renovated and reopened in April.
The Bursaspor football club issued a statement saying the explosion happened "close to the away team stands where our supporters were. We have received the news that around five minutes before the explosion took place, our supporters left the premises."
Christopher James, a freelance writer and teacher living in Istanbul, told CNN he was at a hotel not far from the arena when the blasts occurred.
"We could hear and see the boom, and then after the boom the sound came back towards us," he said. "It sounded like gunshots reverberating and then my phone started buzzing like crazy."
Ramazan Hakki Oztan, a historian from Istanbul who was attending a casual gathering near the arena, also saw the explosions.
"We were at this hotel with this nice view of the old city by Taksim Square," he said. "We saw this huge explosion that happened by the stadium ... and 10 seconds or 15 seconds after there was another explosion. ... The second bomb was smaller in size."
He said he was near the arena earlier in the day and noticed a heavy police presence.
"I think they targeted the cops that were out there by the stadium who were protecting the spectators," he said.
Who is the TAK?
In its statement claiming responsibility, the TAK said two of its members died in the attack.
The TAK is not technically a part of Turkey's main bete noire of Kurdish terrorism, the PKK, but some analysts say that's partly a matter of plausible deniability.
They say the TAK follows the broad direction of the PKK, adhering to its ceasefires and mimicking its targeting, but does not seek approval from the PKK's hierarchy for each attack.
The PKK's critics say that this loose structure allows the TAK to keep its acts of carnage at an acceptable distance from the larger militant group.
But the fact that a faction linked, however loosely, to the PKK has claimed this savage attack will still complicate US relations with Turkey.
The United States arms and assists Kurds fighting in northern Syria against ISIS. But Turkey insists these militants in northern Syria, the YPG, are essentially the PKK in different uniforms.
The US disputes that and lists the PKK as a terror group.
A violent year
This has been a violent year in Turkey. Both ISIS and the Kurdistan Workers Party (PKK) have staged attacks, and the country is still reeling from a bloody and failed attempt at a military coup
Erdogan declared a state of emergency following the coup attempt and authorities carried out a large number of arrests.
ISIS is suspected in a June attack at Istanbul's Ataturk Airport that left 44 people dead and an explosion at an August wedding in Gaziantep, not far from the border with Syria, that killed at least 54 people.
Meanwhile, Turkish security forces continue to clash on a almost daily basis with PKK militants, mostly in predominantly Kurdish parts of southeastern Turkey.
The Turkish army suspects the PKK was behind a September car bombing that killed at least 18 people in the same part of the country. Other attacks have targeted Turkish police and army assets.
Beyond those bombings, the Iraqi and Syrian conflicts have trickled into Turkey, leading to a surge in violence as the nation plays has played host to millions of Syrian and Iraqi refugees.
The US government two months ago ordered all civilian family members of its Istanbul consulate staff to leave Turkey because of increasing threats from terrorist organizations.
This past spring the Pentagon ordered family members to leave Incirlik Airbase in southeast Turkey and the State Department ordered families of employees of the US consulate in Adana to evacuate.
In an address on Saturday, Erdogan said, "It does not matter what is the name and the method of the terror organization who conducted the terror attack. Whenever Turkey takes a positive step towards the future the answer comes as blood savagery and chaos."
During his statement, he called out ISIS, PKK and a movement affiliated with US-based cleric Fethullah Gulen for targeting Turkey. Erdogan has accused Gulen's movement of being involved in the failed coup.
NATO Secretary General Jens Stoltenberg
also condemned Saturday's "horrific acts of terror" in Istanbul. "My thoughts are with those who lost their loved ones in the bomb attacks, with those wounded and with the people of Turkey. We stand united in solidarity with our ally Turkey. We remain determined to fight terrorism in all its forms," he said.
UEFA, European soccer's governing body, said on its verified Twitter account: "UEFA would like to express its deepest condolences to the families of all the victims of tonight's bombing in Istanbul. We strongly condemn this horrible act and send our support to the Turkish Football Federation, Besiktas and Bursaspor football clubs."