A demonstration project of up to six turbines will be built in the water between Orkney and the Scottish mainland.
Energy Minister Fergus Ewing said: "Today we have granted consent to MeyGen Limited to develop the largest tidal turbine array in Europe and the first commercial project off these shores.
"This is a major step forward for Scotland's marine renewable energy industry. When fully operational, the 86 megawatt array could generate enough electricity to power the equivalent of 42,000 homes - around 40% of homes in the Highlands. This exciting development in the waters around Orkney is just the first phase for a site that could eventually yield up to 398MW."
Speaking before the Scottish Renewables Marine Conference, Mr Ewing said developers Aquamarine Power and Pelamis Wave Power are to share a slice of a £13 million wave "first array" support programme, part of the Scottish Government's Marine Renewables Commercialisation Fund.
Mr Ewing said: "We must tackle climate change. We need to reduce our reliance on fossil fuels through better and more efficient uses of energy. Marine energy - a home-grown technology with huge potential - is part of the solution."
The Carbon Trust has estimated wave and tidal resources could provide 20% of the UK's electricity.
Lang Banks, director of environmental group WWF Scotland, said: "This is a significant announcement and a major boost for the marine renewable industry in Scotland."