Being much, much, much, much older than most of you, my favorite gangster flicks go back to classic Hollywood:
1. Little Caesar (1930) - Edward G. Robinson
2. Scarface: The Shame of a Nation (1932) - Paul Muni and George Raft
3. The Public Enemy (1931) - James Cagney
These movies were made when the people they were patterned after were still alive. It took courage to even make them. And although they're dated now, they are the real beginning of the gangster genre.
Cagney teamed with Humphrey Bogart in three movies that are also classic:
1. The Roaring 20s (1939)
2. They Drive by Night (1940)
3. High Sierra (1941) - This one has Bogart as a softer character, but it's still vintage Bogie.
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And yes, the crime/gangster genre is one of my favorites from back in the Golden Days of Hollywood.