Act 1
Scene 1: A calm sea by the Sicilian shore, in sight of Mount Etna
The chorus sings of two goddesses (Vien bell'Aurora). Galatea and Calipso lament that they have fallen in love with mortals (Vo presagendo). As the chorus praises love (Febo tu ancora) Calipso leaves.

Scene 2: The same
Galatea's attempted exit is interrupted by Polifemo. He desires Galatea and wonders why she is not impressed with his pedigree (he is the son of Neptune) or his superhuman strength. Galatea states that she can not love him. Polifemo responds that the flames in his heart are greater than those of Mount Etna (M'accendi 'n sen col guardo). Knowing of his tendency to rage and fury, Galatea wishes that he would understand she has fallen in love with the mortal Aci (Se al campo e al rio soggiorna).

Scene 3: The ships of Ulisse are seen in the distance; Ulisse disembarks with his entourage from one of the ships and is met by Aci.
After a long and arduous journey, Ulisse is happy to have landed. He sees a cave that will serve as suitable accommodation. Aci warns Ulisse of Polifemo the giant cyclops who terrorizes people, killing and devouring them. Aci explains that he knows the giant's routine and has been able to avoid him. He urges Ulisse to leave. But Ulisse wants to see the giant first; he explains his lack of fear in that "The greatest dangers are but common objects." (Core avvezzo al furore dell'armi). Aci muses that Ulisse's bravery is the kind that can vanquish Polifemo. He then sees Galatea, his love, approaching in a ship. (Dolci fresche).

Scene 4: Another part of the shore with cottages belonging to fishermen.
Calipso in disguise meets Ulisse. He is taken with her beauty, while she assures him safety (Vedrai che veglia il cielo).

Scene 5
Polifemo enters and is amused to find Ulisse ready to wage battle. Recognizing his heroism, Polifemo pledges to protect Ulisse and his men, which Ulisse warily accepts but remains on guard (Fa ch'io ti provi ancora).

Scene 6: A grove
In a grove, Galatea is happy to be with Aci but wonders at Aci's lack of concern about Polifemo. Aci responds that he's not afraid of the giant. Galatea promises to visit the grove again in order to meet but must leave because of the approach of Polifemo. Aci responds that her presence gives him joy (Morirei del partir nel momento). He leaves and Galatea wonders what the other sea nymphs will think of her affair with a mortal although she loves him. (Ascoltar no non ti volgio).

Act 2
Scene 1
Calipso muses on the implications of her love to a mortal.
(In the first version: Nerea encourages Calipso to use her charms in mollifying Polifemo in order to liberate Ulisse from the cave where he and his men they are protecting themselves (Un beltà che fa).)

Scene 2
Ulisse approaches with a flock and tells Calipso that Polifemo has given him a shepherd's tasks, and that his men are held captive in a cave. He explains that they are captives until Polifemo's slaves return with presents. If the slaves have brought no presents to Polifemo, then he will devour them. Calipso tells him not to worry because the gods are on his side (Lascia fra tanti malì) (early version:Nel rigor d'avversa stella) Ulisse remarks on her kindness (Fortunate pecorelle!).

Scene 3
Aci revels in his infatuation for Galatea (Lontan dal solo e caro...Lusingato dalla speme) (First version: Zeffiro lusinghier).

Scene 4: A view of the sea
Preparing to meet Polifemo, Galatea sails in her seashell, encouraging the breezes to bring her to Ulisse (Placidetti zeffiretti). Polifemo intercepts her and questions why she would prefer a young boy to his attributes. She refuses him and Polifemo swears revenge on Aci. Galatea continues entreating the breezes to bring her to Aci (Placidetti zeffiretti).

Scene 5: Aci and Galatea
Aci encourages the cupids to bring Galatea safely to shore (Amoretti vezzosetti). Aci and Galatea have an excited exchange in which they reveal their love for one another. Galatea tells Aci to meet her later in a grotto, and Aci promises to do so, his passion for Galatea overcoming his fears (Nell'attendere il mio bene). Galatea is smitten by Aci and wishes that all her hopes are true (Fidati all speranza). (Earlier version: Dal guardo che incatena.)

Scene 6:
Ulisse awakes to find Calipso. Calipso explains who she is to him. She promises him safety if he willll give his heart to her, and alerts him that the slaves are bringing presents to Polifemo. Overjoyed Ulisse sings (Dell'immortal bellezza).

Scene 7: A grove
Aci and Galatea express their love despite foreboding feelings of fear (Tacito movi e tardo).

Act 3
Scene 1: A rock near Mount Etna, at the foot of which in a shady bower is Aci and Galatea.
Polifemo admonishes Galatea (Fugace Galatea). He would have gladly been a water nymph to be with her. But he is more powerful than Jove and will bring about vengeance because of her refusal. He throws the rock and kills Aci.

Scene 2
Galatea mourns Aci (Smanie d'Affanno).

Scene 3: Polifemo's cave
Ulisse and Calipso prepare to meet Polifemo. Ulisse wonders why does she help him. Calipso explains that his previous heroic actions has moved her to help him. She becomes invisible as Polifemo enters, exultant over having exacted revenge. Ulisse offers him some wine from Mount Etna (D'un disprezzato amor). Polifemo drinks and falls asleep. Ulisse takes a burning brand and sticks it in Polifemo's eye and exults in having overcome him (Quel vasto, quel fiero). Calipso rejoices in seeing Ulisse beat Polifemo.

Scene 4: The rock which fell upon Aci
Galatea is happy that Polifemo has been vanquished but implores Jove to restore Aci to life.

Scene 5
The rock opens and a stream springs forth. Aci, now the god of the stream, holds an urn. Both Aci and Galatea thank Jove for restoring his life (Alto Giove).

Scene 6
Now blind, Polifemo wanders the island aimlessly (Furie ce mi strazjate). Aci tells Polifemo that Jove has exacted revenge for killing him (Senti il fato). Polifemo acknowledges that he is consumed with rage.

Scene 7
Ulisse praises the nymphs and all around him (Intessiete ghirlande). All sing a chorus to love (Accendi nuova face).