Here's a nice history of computing from Stanford University.
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Stanley Fifer reports that the first semi-automatic mechanical analog computer was built in England by the Manchester firm of Metropolitan Vickers prior to 1930 (Fifer [1961], p. 29); however, I have so far been unable to verify this claim. In 1931, Vannevar Bush, working at MIT, built the differential analyser, the first large-scale automatic general-purpose mechanical analog computer. Bush's design was based on the wheel and disc integrator. Soon copies of his machine were in use around the world (including, at Cambridge and Manchester Universities in England, differential analysers built out of kit-set Meccano, the once popular engineering toy).
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The first electronic stored-program digital computer to be proposed in the U.S. was the EDVAC (1945).
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The first fully functioning electronic digital computer to be built in the U.S. was ENIAC, constructed at the Moore School of Electrical Engineering, University of Pennsylvania, for the Army Ordnance Department, by J. Presper Eckert and John Mauchly (Completed in 1945).
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There were also others that predate EDVAC and ENIAC, including the Atanasoff-Berry Computer (1939) and Colossus (1944).