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The fighting is worst in the capital, Budapest. Estimates of the number of dead vary from 2,000 to 5,000. Bodies still lie in the streets where they have fallen in what is already being called the Hungarian October Revolution.
The rebellion is reported to have spread into the eastern half of the country and towns like Debrecen, Miskolc, Szeged and Tatabanya.
Mr Nagy was dismissed last year for his liberal policies, but has since been re-admitted to the Hungarian Workers' Party.
Two days ago, the Central Committee of the Communist Party reinstated him as prime minister in an attempt to pacify the Hungarian protesters and end the uprising.
He has promised to meet the demonstrators' demands for economic reforms and the withdrawal of Soviet troops and tanks.
The trouble began with a rally in the capital, Budapest on Tuesday (23 October). About 100,000 students and workers took to the streets of the capital carrying pro-democracy banners.
Police moved in and began firing at the crowds. When they refused to disperse, Erno Gero, First Secretary of the Hungarian Workers' Party, ordered Soviet tanks onto the streets.
Reports received by the British Foreign Office say the tanks fired into the crowds at point blank range, causing hundreds of casualties. But the unarmed protesters refused to give way and eventually the tanks were forced to pull back.
The appointment in July of hardline Mr Gero increased tension in a country desperate to breakaway from its Stalinist past and adopt its own path to socialism.
One of the first western journalists to visit Budapest since the uprising has said clashes are continuing in the working-class district of Csepel and Soroksar. The Soviets are shooting into houses, the journalist said.
He said the Soviet headquarters in the Hotel Astoria were "all shot up".
Electricity, gas and water supplies are still functioning. A ban on alcohol is strictly enforced and people are queuing for bread.
Mr Nagy has issued an amnesty to all rebels who lay down their arms by 2200 local time. He has urged workers and party members to "liquidate all enemies" found with weapons after that hour.
Regular appeals for quiet and order are being broadcast. People have been warned - on pain of being shot - against being in the streets of the city between 1100 and 1600 local time.
Reports on 30 October said Soviet troops were beginning to pull out of the Hungarian capital and negotiations would begin shortly for their complete withdrawal from Hungary.
Mr Nagy promised free elections and a return to the multi-party system. He also announced Hungary's withdrawal from the Warsaw Pact.
However, he was betrayed by Janos Kadar and the Soviet tanks returned on 4 November. Mr Nagy was overthrown, put on trial and executed in 1958.
Mr Kadar took control and ruled for the next 30 years.
He began by waging war on what he saw as the "counter-revolutionaries" and up to 200,000 Hungarians fled.
Later he adopted a more relaxed approach and allowed some private initiative with the result Hungary became one of the only Communist countries self-sufficient in food. However, industry suffered and eventually led to the creation of an opposition movement and the ousting of Mr Kadar in 1989.
That same year, 31 years after his execution, Mr Nagy was declared the victim of a show trial and his body officially reburied with full honours.
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