May

• May 1
  • Bob Askin replaces Jack Renshaw as Premier of New South Wales.
  • The Battle of Dong-Yin occurs as a conflict between Taiwan and the People's Republic of China.
  • Liverpool wins the FA Cup Final, beating Leeds Utd 2–1.
• May 5 – Forty men burn their draft cards at the University of California, Berkeley, and a coffin is marched to the Berkeley Draft Board.
• May 6 – A tornado outbreak near the Twin Cities in Minnesota kills 13 and injures 683.
• May 7 – The U.S. Steel freighter SS Cedarville collides with the SS Topdalsfjord and sinks near the Mackinac Bridge, killing 25. 10 are rescued from the Cedarville, the 3rd largest lake ship to sink after its sister the SS Carl D. Bradley, and the SS Edmund Fitzgerald.
• May 9 – Pianist Vladimir Horowitz returns to the stage after a 12-year absence, performing a legendary concert in Carnegie Hall in New York.
• May 12
  • West Germany and Israel establish diplomatic relations.
  • The Italian liner SS Michelangelo enters service.
• May 13 – A West German court of appeals condemns the behavior of ex-defense minister Franz Josef Strauss during the Spiegel scandal.
• May 21 – The largest antiwar teach-in to date begins at Berkeley, California, attended by 30,000.
• May 22 –
  • Several hundred Vietnam War protesters in Berkeley, California, march to the Draft Board again to burn 19 more cards. Lyndon Johnson is hung in effigy.
  • The first skateboarding championship is held.
• May 25 – Muhammad Ali knocks out Sonny Liston in the first round of their championship rematch with the "Phantom Punch" at the Central Maine Civic Center in Lewiston.
• May 27 – Internazionale beats Benfica 1-0 at the San Siro, Milan and wins the 1964-65 European Cup (football).
• May 29 – A mining accident in Dhanbad, India kills 274.
• May 31 – Racing driver Jim Clark wins the Indianapolis 500, and later wins the Formula One world driving championship in the same year.