He was the son of John Bancroft & Mary Curwen and baptised on 9th September 1544 at Farnworth in Lancashire. His early education was at Farnworth Grammar School, and later at Cambridge, firstly at Christ's College & then at Jesus College.
He was ordained in 1570 as chaplin to the Bishop of Ely, and then became on of the preachers at the University. In 1584 he was made rector of St Andrew's in Holborn and a year later in 1585 was appointed treasurer of St Paul's Cathedral.
On 9th February 1589 he preached at St Paul's Cross and the theme of his sermon was a passionate attack on the Puritans and is said to have "denounced the exercise of the right of private judgement, and set forth the devine right of bishops in such strong language that one of the Queen's councillors held it to amount to a threat against the supremacy of the crown"'
In June 1597 he was consecrated Bishop of London, and from that time because of the age and incapacity of Archbishop Whitgift, he was virtually invested with the power of primate and had the sole management of ecclesiastical affairs.
Queen Elizabeth died in 1603, and he was present at her death.
In 1604 Archbishop Whitgift died and he was eventually appointed his successor, albeit it after strong opposition from parliament because of a Book of Canons which he had produced for royal approval earlier in the year.
Probably his most important fact in his life was his appointment by King James to oversee the translating of a new authorised version of the Bible, which was eventually published in 1611.
Sadly Richard never saw the completion of his work on the Bible project because he died at Lambeth Palace on 2nd November 1610. In the preface of the new King James Version of the Bible, the translator refer to him as "chief overseer & task-master under his Majesty, to whom were not only we, but also our whole Church,much bound"