Bowen was educated at Cardigan County School, University College, Aberystwyth, St John's College, Cambridge, Brussels and the Inns of Court. He served in the Army for five years during World War II, reaching the rank of Captain. He was a barrister and became a King's Counsel (KC) in 1952.
At the 1945 general election, Bowen was elected at the first attempt as Member of Parliament for Cardiganshire, succeeding Owen Evans. With only a Labour opponent he comfortably held the seat. He was re-elected in the 1950 general election with the largest Liberal majority at that election, despite facing both Conservative and Labour opposition. During most of the 1950s he was one of a tiny group of only five or six Liberal MPs left at Westminster. He failed to succeed Clement Davies as leader of the Liberal Party in 1956 when Jo Grimond was chosen. His seat remained safe until the 1964 election when Labour cut his majority to around 2,000. At the 1966 general election, he was defeated by the Labour candidate by just 523 votes.
Bowen's political career came second to his judicial career although he was a President of the Liberal Party and member of its council. He was Recorder of Merthyr Tydfil and of Carmarthen. After the 1964 election when Labour's majority in the House of Commons was just four seats, he was persuaded by the Government to accept the position of Deputy Speaker. This put the Labour majority in the House up to five.
Politically, he was on the right of the Liberal Party. During the Suez Crisis, he defended the Conservative Government against opposition "carping criticisms". One of the chief opposition critics of the Government over Suez was his own party leader, Jo Grimond.
He served as National Insurance Commissioner for Wales, 1967-86, and as president of St Davids University College, Lampeter, 1977-92. One of the libraries and a research centre at Lampeter carry his name. In 1971-72 he chaired a governmental committee set up to examine road signs policy in Wales.