The two Nordic states were expected to be accepted as members of the defense alliance quickly after they announced their interest in joining NATO. Turkey, one of the group's most important and powerful members, is not happy about joining NATO.
The reasons are complicated, emotional and steeped in decades of violent history.
In the wake of Russia's invasion of Ukraine, Sweden and Finland decided to abandon their nonaligned position and join NATO.
The idea that the Nordic states might join the alliance made Moscow bristle. NATO expansion is something it has previously cited to justify invading Ukraine.
Turkey's President has the power to determine the future of the NATO alliance and its size in the face of Russia's war.
Turkey has already blocked an early attempt by NATO to fast-track the applications of Sweden and Finland, saying their membership would make the alliance a place where representatives of terrorist organizations are concentrated.As of 2022, NATO has expanded to let in three former Soviet states and all of the former Warsaw Pact countries.
The clash has sent Western diplomats scrambling to bring Turkey on side, as Ankara presented a list of grievances to NATO ambassadors about its issues with the Nordic states.
The PKK is a Kurdish Marxist movement that has been fighting Turkish forces on-and-off since 1980. It operates mostly in Turkey and northern Iraq.
The PKK is classified as a terrorist organization by Turkey, as well as by the US, Canada, Australia and the European Union.
Sweden was one of the first countries to designate the group as a terrorist organization.
Turkey says that Sweden provides protection for PKK members. Sweden says it supports other Kurds who are not in the PKK, but the details are more complicated.
When contacted by CNBC, Sweden's foreign ministry declined to comment.
According to the Crisis Group, between 30,000 and 40,000 people have died in fighting between the PKK and Turkish government. The PKK has carried out many attacks in Turkey.Members of Turkish Armed Forces (TSK) continue operations against the PKK, listed as a terrorist organization by Turkey, the U.S. and the EU, and the Syrian Kurdish YPG militia, which Turkey regards as a terror group, within Turkey’s Operation Peace Spring in Ras Al Ayn, Syria on October 17, 2019.
Turkey's opposition to joining NATO appears to be related to its smaller Kurdish population than Sweden, but its foreign policy tends to be the same.
Sweden and other EU countries halted weapons sales to Turkey in the middle of the year due to Ankara's military action against Kurdish groups in Syria.
Turkey wants Sweden to extradite a list of people they have charged with terrorism. He wants Sweden and Finland to publicly disassociate themselves from the PKK and its affiliates.
The Turkish perspective is very simple for Hakki Akil, a former Turkish ambassador.
If Sweden and Finland want to join a security alliance, they have to give up their support for the PKK. They have to accept Turkish requests of 30 terrorists, which are very specific cases.
An estimated 30 million people are Kurds, which is the largest ethnic group without a homeland. Mostly Sunni Muslim, they have their own language and customs.
Kurds make up 20% of Turkey's 84 million strong population, with some Kurds holding important positions in Turkish politics and society.
Saddam Hussein's chemical gas attacks on Kurds in Iraq in the late 1980s were one of the many examples of genocide they have been victims of. Various Kurdish groups have pushed for Kurdish independence and statehood over the years, some peacefully and some through violence.Kurds celebrate to show their support for the independence referendum in Duhok, Iraq, September 26, 2017.
Kurdish fighters in Syria linked to the PKK received weapons support and funding from the U.S and Europe. Turkey launched attacks on the Kurds in Syria after this.
Muhammet Kocak, an international relations specialist based in Ankara, told CNBC that people who have been fighting with Turkey for 40 years have killed tens of thousands of civilians.
Turkey is not happy about the fact that they became good guys because of their work against the Islamic State.
Western governments hailed the Kurdish fighters as allies, and several EU countries put various embargoes on Turkey due to their targeting of Kurdish militias in Syria, highlighting the intractable differences between how each side perceived the fighters.
Hussein Ibish, a senior resident scholar at the Arab Gulf States Institute in Washington, says that the tension between Turkey and Sweden is related to how each country defines terrorist.
It's not just a matter of Sweden's liberal policies towards Kurdish refugees. Ibish said that it was also a reflection of differing definitions of intolerable Kurdish extremism.
Turkey considers all Kurdish groups to be PKK front organizations. There are many non-PKK Kurdish entities and organizations in and from Turkey, as well as the Western-backed Syrian Democratic Forces (SDF) in Syria and a number of Iraqi Kurdish groups.
Kurdish refugees and asylum seekers have been accepted by Sweden for a long time. Kurds have seats in Sweden's Parliament.
Most of the Kurds living in Sweden have no affiliation with the PKK, but the Swedish government has supported members of other Kurdish organizations.
Turkey says the PKK and PYD are the same, but Sweden says they are different.
The Syrian Democratic Council (SDC), the political wing of the Kurdish-led militia group the Syrian Democratic Force, is supported by the Swedish government. PKK terrorists are the dominant group in the SDC, according to Ankara.
Sweden's government announced an increase in funding for Kurdish groups in Syria to $376 million by 2023, saying it remained an active partner.
With Swedish elections coming up in September, it's unlikely that the government will make any major concessions that would make it look weak.
Others believe that Erdogan will not block Sweden and Finland from joining NATO, but that he is trying to improve his popularity at home.
If Turkey can get concessions from the Western powers and its NATO allies, it will not try to block Sweden from joining the organization, according to the Arab Gulf States Institute.
The Russian invasion of Ukraine has reminded many Turks of the value of membership in NATO, as the war now focuses on parts of that country that are adjacent to Turkey.
NATO may face a stalemate for some time if Turkey doesn't get what it wants.