The 37-year-old captained India in five Tests, 51 one-day internationals and 155 Twenty20 internationals, leading the side to their first-ever 50-over World Cup triumph on home soil last year.
Under Kaur, India also won two T20 Asian Cup titles and two Asian Games gold medals, most recently last month at the Aichi-Nagoya Games.
"Today, I step down as captain of the Indian women's cricket team across all formats," Kaur wrote on social media.
"This is not an easy decision. It carries years of memories, emotions, sacrifices, and moments that will stay with me for the rest of my life."
Indian media said Kaur's decision came a day after she met Devajit Saikia, secretary of the Indian cricket board (BCCI), to discuss the future of the women's team.
Kaur, who became this year the highest-capped player in women's international cricket, said she intended to keep on playing.
"I gave this team everything I had," she added.
"I carried the responsibility with pride, and I will always be grateful for the privilege of leading Indian women's cricket.
"Now, I'm ready to play for the love of the game again. Captaincy ends. The journey continues."
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