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He began acting with London's politically minded Unity Theatre. In 1956 Landis had an important supporting role as a Cockney soldier in A Hill in Korea. Michael Caine, who made an unnoticed screen debut in the film, recalled in his autobiography that great things were expected of Landis at the time, but didn't materialise.
Indeed, Landis' other film roles have been pretty minor, as in Dunkirk (1958), Doctor in Distress (uncredited, 1963), The Informers (uncredited, 1963), Private Potter (1962), Operation Bullshine (1959) and Ransom (1975) with Sean Connery.
However Landis kept active in other media, returning to the Unity Theatre to help out the largely amateur casts, and occasionally directing there, such as a well-praised late '60s production of Death of a Salesman. On television he appeared in Dixon of Dock Green (1955-1976), The Avengers (1964), The Indian Tales of Rudyard Kipling (1964), The Long Street (1965), The Saint (1966), Locate and Destroy (1966), Irish-Greek, Man In a Suitcase (1967), Jason King (1971), Angels (1975), Z-Cars (1968-1974) and Law and Order (1978). He also appeared in one of Jeremy Brett's last appearances as Sherlock Holmes in The Memoirs of Sherlock Holmes (1994) among many others.
More characteristically, he was cast in Jack Rosenthal's one-man show Bar Mitzvah Boy (BBC, 1976); played one of Arthur Daley's crooked cronies in Minder (1982), and has had roles in Bergerac (1987); Howards' Way (1985); You Rang, M'Lord? (1990) and Jeeves and Wooster (1991).
He is more recently remembered as being a regular in the BBC soap opera EastEnders (1995-1997) where he played the friendly barber, Felix Kawalski, but Felix was killed off in 2005, off-screen. He continues to act and has made recent appearances in the ITV police drama The Bill (2003), Casualty (2004), BBC comedy My Family (2004) BBC hospital drama Holby City (2006) and most recently the Channel 4 sitcom Friday Night Dinner (2012).
Landis was elected as President of Equity, the British actors' union, in July 2002.