|Ab urbe condita||1416|
|Balinese saka calendar||584–585|
|Chinese calendar||壬戌年 (Water Dog)|
3359 or 3299
— to —
癸亥年 (Water Pig)
3360 or 3300
|- Vikram Samvat||719–720|
|- Shaka Samvat||584–585|
|- Kali Yuga||3763–3764|
|Minguo calendar||1249 before ROC|
|Seleucid era||974/975 AG|
|Thai solar calendar||1205–1206|
789 or 408 or −364
— to —
790 or 409 or −363
Year 663 (DCLXIII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. The denomination 663 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.
- Emperor Constans II launches an assault against the Duchy of Benevento (Southern Italy). Taking advantage of the fact that Lombard king Grimoald I is engaged against Frankish forces from Neustria, Constans disembarks at Taranto and besieges Lucera and Benevento.
- Constans II visits Rome for twelve days—the only emperor to set foot in Rome for two centuries—and is received with great honor by Pope Vitalian. Constans gives the order to strip buildings, including the Pantheon, of their ornaments, which will be carried back to Constantinople.
- Constans II moves the imperial court from Constantinople to Syracuse. He tries to stop the Arab conquest of Sicily, and restores Rome as seat of the Byzantine Empire. Constans strips sacred altar vessels from churches all over Rome.
- May 8 – Battle of Forino: The Byzantine army, led by Constans II, is defeated by the Lombards under Romuald I. He seizes Taranto and Brindisi, receiving military aid from the Bulgar Alcek horde, who are settled in the area of Ravenna.
- King Oswiu of Northumbria invades Pictland (modern Scotland). He establishes overlordship of, at least, the Southern Pictish sub-kingdoms of Fortriu and Fib (and possibly Circinn).
- A brief outbreak of plague hits Britain (approximate date).
- June 5 – In China, the Daming Palace becomes the government seat and royal residence of Emperor Gao Zong of the Tang Dynasty.
- Battle of Baekgang: Korean Baekje forces and their Japanese allies are defeated in a naval battle, by a joint Silla–Tang coalition.
- Nasr ibn Sayyar, Arab general (d. 748)
- Ōtsu, Japanese prince and poet (d. 686)
- Song Jing, chancellor of the Tang Dynasty (d. 737)
- Yamanobe, Japanese princess (approximate date)
- Zhang Yue, chancellor of the Tang Dynasty (d. 730)
- Ago, duke of Friuli (approximate date)
- Cunibert, bishop of Cologne (approximate date)
- Gartnait IV, king of the Picts (approximate date)
- Guaire Aidne mac Colmáin, king of Connacht (Ireland)
- Gwisil Boksin, Korean general of Baekje (Korea)
- Fryde, et al. "Handbook of British Chronology", p. 223