Crockett was educated at Pontypridd Grammar School  and King's College London before beginning his ordained ministry. He was ordained in 1971 and was a curate at Aberdare and Whitchurch (Cardiff). From 1978 to 1986 he was Vicar of Llanafan y Trawsgoed then Rector of Dowlai before eight years as the Secretary of the Board of Ministry of the Church in Wales. A fluent Welsh speaker, his final post before his ordination to the episcopate was as Archdeacon of Carmarthen (1999-2004).
Crockett was the first divorced person in the United Kingdom to be appointed as a bishop in a mainstream church, amidst some controversy. In 1985 he separated from his first wife, with whom he had three children, and remarried Caroline Ann Owen (born 1959) in 1999. He ordained his second wife in 2006.
- Who's Who 2008: London: A & C Black, 2008 ISBN 978-0-7136-8555-8
- Debrett's People of Today. London: Debrett's, 2008, ISBN 978-1-870520-95-9
- BBC Wales obituary
- Crockford's Clerical Directory 2008/2009 (100th edition), Church House Publishing, ISBN 978-0-7151-1030-0)
- - 'The Right Rev Anthony Crockett: Forthright Bishop of Bangor' - The Independent 11 July 2008
- "First divorced bishop appointed". BBC News. Retrieved 2012-08-30.
- 'The Right Reverend Anthony Crockett' - The Daily Telegraph 1 July 2008
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