Thyestes felt he had been deprived of the Mycenean throne unfairly by his brother, Atreus. The two battled back and forth several times. In addition, Thyestes had an affair with Atreus' wife, Aerope. In revenge, Atreus killed Thyestes' sons and served them to him unknowingly. After eating his own sons' corpses, Thyestes asked an oracle how best to gain revenge. The advice was to father a son with his own daughter, Pelopia, and that son would kill Atreus.
When Aegisthus was born, his mother was ashamed of her incestuous act. She abandoned him and he was raised by shepherds and suckled by a goat. Atreus, not knowing the baby's origin, took Aegisthus in and raised him as his own. When Aegisthus reached adulthood, Thyestes revealed his true parentage, that he was both father and grandfather to Aegisthus, who then killed Atreus and seized the throne.
Aegisthus and Thyestes ruled over Mycenae jointly, exiling Atreus' sons, Agamemnon and Menelaus to Sparta, where King Tyndareus gave the pair his daughters, Clytemnestra and Helen, to take as wives. At his death, Tyndareus gave his throne to Menelaus, who then helped Agamemnon overthrow Aegisthus and Thyestes.
After Agamemnon left Mycenae for the Trojan War, Aegisthus wanted to seduce his wife, Clytemnestra (mother of Erigone). Agamemnon had left Clytemnestra with a singer; as long as the singer was present, Clytemnestra resisted Aegisthus. Aegisthus then took the singer to a deserted island, and Clytemnestra was seduced. On the kings' return after the ten-year war, Aegisthus helped Clytemnestra kill Agamemnon (and his new concubine, Cassandra); they subsequently ruled Mycenae for seven years.
Homer, Od. iii. 263, iv. 517; Hyginus, Fab. 87.