Do Muslims believe in original sin?

Adam's mistake was forgiven

The Quran tells the story of Adam and Eve eating from the forbidden tree, but it ends with mercy rather than a curse on humanity: "Then Adam received from his Lord [some] words, and He accepted his repentance" (Quran 2:37). Another verse adds: "Then his Lord chose him and turned to him in forgiveness and guided [him]" (Quran 20:122).

In Islam, the slate was wiped clean. Adam went on to become the first prophet, and the Quran blames both Adam and Eve together, not Eve alone.

No one carries another's sin

A principle repeated several times in the Quran is personal responsibility:

The same idea appears in Quran 6:164. You won't be judged for what your parents, ancestors or Adam did, only for what you did yourself.

Born pure, on the fitrah

Instead of being born in sin, Islam teaches that people are born on the fitrah, a natural, clean state that recognises God. The Prophet ﷺ said: "Every child is born with a true faith of Islam (i.e. to worship none but Allah Alone) and his parents convert him to Judaism or Christianity or Magianism" (Sahih al-Bukhari 1385).

That's why Muslims believe children who die young are not punished. See What happens to babies who die in Islam?

Then how are people saved?

Without original sin, there's no need for someone else to pay for it. Islam teaches that people reach God's forgiveness through faith, sincere repentance and good deeds, all by His mercy. There's no baptism needed to remove an inherited stain.

This is good news for converts: when you become Muslim, your past sins are forgiven and you start fresh. See Are all my sins forgiven when I become Muslim?

And when you slip after that, the door of repentance stays open. See Will Allah forgive me if I keep committing the same sin?