He was born in Breslau, where his father was a professor of political economy. He studied in that town, Bonn, and in Berlin, 1842-1845, busying himself especially with literature and Sanskrit archaeology. He received a doctor's degree at Breslau. In 1846, he visited England and France in connection with his studies. On his return to Germany, he went to the University of Berlin, where he was privatdocent, and in 1856 became an adjunct professor of the language and literature of ancient India. In 1867 he was made full professor. He was a member of the Academy of Sciences of Berlin, and was the author of many books and periodical contributions on classical subjects. He was a close friend of Max Müller.
Weber died in Berlin.
- Indische Studien, 1849–85
- Weiße Jadschurveda, London 1849-1859 (3 Bde.)
- Schwarze Jadschurveda, Leipzig 1871-1872
- Tscharanawyuha. Übersicht über die Schulen der Vedas, Berlin 1855
- Akademische Vorlesungen über indische Litteraturgeschichte, Berlin, 1852; 2d ed. 1876 (translated by Zachariae and Mann, London, 1878)
- a translation of Kalidasa's drama Mālarikā und Agnimitra, 1856
- Indische Skizzen, Berlin 1857
- Indische Streifen, Berlin 1868-1879 (3 vols.)
- Verzeichnis der Sanskrithandschriften der königlichen Bibliothek zu Berlin, Berlin 1853-1892
- Über das Catrunjaya des Mahâtmyam, Leipzig 1858
- an edition of Hala's Saptasataka, 1881
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