What are the curses given to Ravana?
In this chapter of the Yuddha Kanda of the Ramayana, right before deciding to wake up Kumbakarna, Ravana enumerates various curses that he's received which have led up to the defeat he is now experiencing:
All that great penance performed by me became a waste indeed, for, I who am equal to Mahendra the Lord of Celestials have been defeated by a mere man! The terrible words of Brahma the Lord of Creation saying, 'You know of the threat from men' appear true. That is so. I sought immunity from death at the hands of celestials, titans or celestials musicians, or semi divine beings or demons or serpent-demons, but of man I made no mention. Rama the son of Dasaratha is the man I think of whom Anaranya born of Ikshvaku dynasty formerly spoke, saying: 'O, the worst of demons, the worst of your race and the wretched one! In my race will be born a man who will slay you in battle with your sons, ministers, army, horses and charioteer. Further, I was formerly cursed by Vedavati when she was humiliated by me. The same Vedavati is born as the highly fortunate Seetha the daughter of Janaka. What was predicted by Parvati the consort of Shiva, Nandishvara the attendant of Shiva, Rambha the wife of Nalakubara and Punjikasthala the daughter of Varunda has come to pass! The words of the sages never prove false.
Let me enumerate them more systematically:
In this chapter of the Uttara Kanda of the Ramayana, Ravana goes to the Sharavana forest, the birthplace of Shiva's son Kartikeya, and he mocks Nandi for having a monkey-like face. So Nandi curses him to be defeated by a race of monkeys:
As O Dasanana, deriding me for my monkey like form, thou hast indulged in a laughter resembling the bursting of thunder, so Vanaras endowed with prowess, and possessed of my form and energy shall be born for compassing the destruction, of thy race. And armed with teeth and claws, and fierce and furnished with the fleetness of thought, and mad for encounter, and bursting with vigor, and resembling moving mountains. And being born, they shall crush thy high pride and power along with thy courtiers and sons.
In the next chapter of the Uttara Kanda of the Ramayana, Ravana attempts to ravish Brihaspati's granddaughter Vedavati, but she commits suicide to prevent that and she curses him right before she dies:
Thou abject, having been outraged by thee I wish not to live. Therefore, O Raksha, I will enter into fire in thy very presence. And as I have in this world been dishonored by thee thou at nefarious. I shall again be born to compass thy destruction. It lieth not in a female to slay a male intent on sin; and if I utter a curse, it shall cost my asceticism. But if I have done anything, given away any thing, offered obtations unto the fire, then I shall be the chaste daughter of some virtous person, albeit unborn of any female vessel.
As I discuss in this answer, Vedavati was reborn as a Sita lookalike and let to Ravana's demise.
In another chapter of the Uttara Kanda of the Ramayana, Ravana kills Anaranya king of Ayodhya, who curses Ravana to be killed by a future descendant of Anaranya:
If I have practised charity, if I have offered oblations into fire, if I have carried on pious, penances, if I have governed my people well, then be my words verified! There shall spring in the line of the high souled Ikshwaku, one named Rama son unto Dasaratha, who shall deprive thee of thy life.
But the ultimate curse which led to Rama killing Ravana is the curse of the Sanatkumaras on Jaya and Vijaya, which I discuss in my answer here.
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