The complaint signed by five lawmakers from the Left and Green parties is the first of its kind in Sweden against a head of state.
The suit relates to the conflict in Turkey's Kurdish majority southeast, which has been battered by renewed fighting between Kurdish rebels and Turkish security forces since a fragile truce collapsed in 2015.
"We are five lawmakers handing in a complaint... (requesting) punishment for genocide, crimes against humanity and war crimes," Annika Lillemets, a MP for the Left party, told a news conference in Stockholm.
A Swedish law adopted in 2014 allows the country's courts to judge cases of alleged crimes against humanity regardless of where they have been committed or by whom.
The law stipulates that "anyone, who in order to completely or partially destroy a national or ethnic group of people" kills, causes serious pain or injury is "guilty of genocide".
The Public Prosecution Offices said it would now decide whether to initiate a preliminary investigation, adding that "it may take a while".
Carl Schlyter, an MP for the Greens, said he hoped other lawmakers in European countries would follow their move.
"If (Erdogan) is hindered from roaming around in Europe and influencing European countries the way he wants, then I hope that this will affect his politics," he said.
The UN Human Rights Office in March released a report on allegations of "massive destruction, killings and numerous other serious human rights violations committed" between July 2015 and December 2016 in Turkey's southeast.
Hundreds of members of the Turkish security forces have been killed since the collapse of the ceasefire, and the army has claimed killing thousands of militants.
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