The Reserve Bank of India was established in the year 1935 in Calcutta, in accordance with the provisions of the Reserve Bank Act of India.

Reuters

The central office of the bank was later moved to Bombay in 1937, and though originally privately owned, it was soon nationalised in 1949.

What you may not know is that the bank also served as Burma's central bank till 1947 and Pakistan's central bank till 1948!