Series with dual-language texts:
Aris & Phillips Classical Texts -- chiefly classical authors; extensive notes, mostly historical and literary.
Loeb Classical Library (available in print and online) -- for mostly classical (with some late antique) works in Greek and Latin. Older works in the series tended to be bowdlerized. Translations are workmanlike rather than literary.
Dumbarton Oaks Medieval Library -- for medieval Latin, Greek, and Old English works.
I Tatti Renaissance Library -- for Renaissance Latin works.
Series with English translations only:
Translated Texts for Historians -- for late antique works in Latin, Greek, and other languages.
Most canonical classical works have been translated for the Oxford World's Classics and Penguin Classics series.
Collection Budé (= Collection des universites de France) -- a wide variety of Latin and Greek texts with extensive notes. Dual-language.
Sources Chrétiennes -- Patristic texts in Greek, Latin, and Syriac with extensive notes. Dual-language.