Answer: The best grape varieties for the Phoenix area are in the European class (Vitis vinifera). Flame seedless grapes and Thompson seedless grapes are both in this class and these plants can be easily found for sale locally. The lower desert summer heat encourages these varieties to produce lots of sugar, creating a wonderfully sweet flavor. Interestingly, the heat also keeps some pigmented grapes from attaining full color, even when they are fully ripe. For example, flame seedless grapes stay green during the hottest months even when they are ready to eat. This requires one to perform taste tests for ripeness before picking them.
For a home gardener, a grape vine can be allowed to sprawl out on a trellis, fence, or wall and over time fruit will be produced. Grape vines need at least a half day of direct sun to grow well and produce. They take the oven like heat of summer afternoons in the Phoenix area very well as long as they have adequate water.
Prune your grapevines back in January or February.
Best wishes with your grapevines.
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