The Bhagavad Gita is part of the Mahabharata, which was written by the sage Vyasa. In addition to being the author, he is also a major character in the Mahabharata, although not the Gita.
It is unclear when exactly the Mahabharata was first written, although it is known that the Gita was a later addition to it and that it did not appear in its present form until around 400 CE. Vyasa's authorship of the epic is also questionable, as the length of the poem is some 100,000 couplets, about seven times the combined length of the Iliad and Odyssey put together. It is more likely that the works of multiple writers were combined into one.
Physicist J. Robert Oppenheimer famously thought of a quote from the Gita, "Now I am become Death, the destroyer of worlds," upon witnessing the first test of a nuclear bomb in 1946.