William and Kate, known as the Earl and Countess of Strathearn when in Scotland, were met by V&A Dundee director Philip Long before entering the £80 million museum for a tour.
The first V&A museum anywhere in the world outside London, the building designed by Japanese architect Kengo Kuma on the banks of the River Tay is the centrepiece of an ongoing £1 billion regeneration of Dundee’s waterfront.
The couple were introduced to Mr Kuma, who talked them through the inspiration for his award-winning design.
William, wearing a navy blue suit, then accompanied his wife at a sit-down with young people who help design activities at the museum.
They were later led into the Scottish Design Galleries where Kate admired a 2015 dress designed by Christopher Kane and viewed Charles Rennie Mackintosh’s Oak Room.
At the end of the tour, the duchess, royal patron of V&A, gave a speech to guests before declaring the museum officially open.
Kate, who was wearing a coat by McQueen in a Black Watch-style design, said: “I have been amazed by the breathtaking architecture and the sheer range of design on display.
“Most of all, I’ve been moved by the passion of the people who work here and the dedication of all those who have helped make this important project succeed.”
She said everyone involved deserved “enormous credit”.
The duchess added: “I hope that this museum will serve as an inspiration to others, showing how great things can be achieved when different communities all pull together. With this thought in mind, I’m pleased to declare V&A Dundee officially open.”
Joined by the duke, she unveiled a plaque before they left the museum to greet members of the public who cheered the couple.
Mr Long said: “V&A Dundee is an ambitious new international museum and we are honoured that the Duke and Duchess of Cambridge today declared the museum officially open.
“V&A Dundee is at the heart of Dundee’s transforming waterfront and is a key part of the city’s plans for an exciting future.
“Within our remarkable building, designed by Kengo Kuma, we are proud to host world-class permanent galleries and remarkable exhibitions that have already delighted and inspired hundreds of thousands of people.”
Lord Lieutenant Ian Borthwick, Lord Provost of Dundee, said: “I am delighted to welcome the Duke and Duchess of Cambridge to V&A Dundee.
“This magnificent building is continuing to raise the city’s global profile and celebrates Scotland’s rich cultural heritage.
“Dundee is a Unesco City of Design, a place where culture is helping to transform people’s lives. This museum has already brought huge benefits to the city of Dundee.”
Later, the couple met employees and their families from a tyre factory which is to cease production.
Michelin announced in November that the plant, which employs 845 workers, would stop production by mid-2020.
The royal couple attended a meeting of the Michelin Action Group, which is aiming to find and take forward solutions for repurposing the plant and providing a sustainable future for its workforce.
Michelin has made a deal with Scottish ministers to repurpose the factory in an effort to secure a future in low carbon transport and manufacturing.
The action group meets regularly and includes representatives from Dundee City Council, the Scottish and UK governments, union officials, Scottish Enterprise and Michelin