Depends on what you consider to be a cell phone. But Wikipedia credits a Motorola researcher to be the inventor of the "first practical mobile phone for hand-held use in a non-vehicle setting", and it was demonstrated in 1973.
In 1979, Japan launched a citywide commercial cell phone network, so it's a good bet that the first commercial cell phone sales would be in 1973 or 1974.
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The first cellular phones were built by Motorola in 1973.
The first cell phone call was on April 3, 1973 by Motorola engineer Martin Cooper. It was another few years before cellular service became widely available, and phones became lightweight enough to carry around.