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Up to 14 people were also injured in simultaneous blasts at a bank and memorial in the capital, Rome.
Another bomb was later discovered by police near Milan's La Scala opera house.
The first explosion happened at about 16.45 local time on the third floor of Milan's Banca dell' Agricoltura (National Agricultural Bank), in Piazza Fontana.
Michelle Carlotto, 27, a clerk at the bank, said: "I was sitting at my desk behind the bank counter. I heard a blast, a bolt which stunned me.
"In the smoke I saw a body fly from the public section above the counter and fall one yard away from me. I was shocked, I couldn't move."
Many of the wounded were taken to hospital where they remain in a serious condition.
Police said eight kilograms (18lbs) of explosives had been placed in an aluminium box and a slow burning fuse was attached to the device.
Within an hour of the blast, at least another 14 people were injured by three further explosions which shook two areas of Rome.
Another bomb was discovered by police a few hours later inside a separate bank in Milan, near the La Scala opera house.
Officers immediately detonated the device in the courtyard of the Banca Commerciale Italiana.
Police in Rome have detained four men so far but no organisation has yet claimed responsibility for the attacks.
The explosions come at a time of deep social unrest and political uncertainty in Italy.
Italian President Giuseppe Saragat condemned the attacks and pledged to "restore the law willed by the people and its sovereignty."
The Prime Minister, Mariano Rumor, called an emergency cabinet meeting and in a news conference he said the explosions were "an act of barbarism which has no precedent in the history of the country."
He later sent a telegram to the Mayor of Milan, Aldo Aniasi, which ordered the Minister of the Interior to "act with the maximum severity against those who want to poison the peace of the Italian people."
In total, 16 people were killed and 58 wounded in what became known as the Piazza Fontana massacre.
Over 4,000 arrests were made in the aftermath of the bombings and one of the suspects, Guiseppe Pinelli, died after falling out of the fourth floor window of the police station where he was being held.
The bombing marked the start of a series of attacks known as "the strategy of tension" between 1969 and 1974 by the right wing group, Ordine Nuovo.
Their aim was to prevent the country falling into the hands of the left -wing by duping the public into believing the bombings were part of a communist insurgency.
The attacks appeared to come to an end in 1974 but the worst strike was to come in 1980, when a suitcase with over 18kg (40lbs) of explosives went off in a train station, in Bologna, killing 85 and wounding more than 200 people.
One of the original suspects, Pietro Valpreda, was eventually tried alongside two neo-Fascists and numerous other secret service agents in 1974, but all the accused were acquitted in 1985 following numerous trials and appeals.
A new trial opened in 1999 after a Milan judge, Guido Salvini, claimed to have discovered the real bombers.
Three neo-fascists, Delfo Zorzi, Carlo Maria Maggi, and Carlo Rognoni were sentenced to life imprisonment but the decision was overturned on appeal in March 2004.
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